In two-prover one-round interactive proof systems, no-signaling provers are those who are allowed to use arbitrary strategies, not limited to local operations, as long as their strategies cannot be used for communication between them. Study of multi-prover interactive proof systems with no-signaling provers is motivated by study of those with provers sharing quantum states. The relation between them is that no-signaling strategies include all the strategies realizable by provers sharing arbitrary entangled quantum states, and more.
This paper shows that two-prover one-round interactive proof systems with no-signaling provers only accept languages in PSPACE. Combined with the protocol for PSPACE by Ito, Kobayashi and Matsumoto (CCC 2009), this implies MIPns(2,1)=PSPACE, where MIPns(2,1) is the class of languages having a two-prover one-round interactive proof system with no-signaling provers. This is proved by constructing a fast parallel algorithm which approximates within an additive error the maximum value of a two-player one-round game achievable by cooperative no-signaling players. The algorithm uses the fast parallel algorithm for the mixed packing and covering problem by Young (FOCS 2001).
Non-signaling proofs, motivated by quantum computation, have found applications in cryptography and hardness of approximation. An important open problem is characterizing the power of no-signaling proofs. It is known that 2-prover no-signaling proofs are characterized by PSPACE, and that no-signaling proofs with $poly(n)$-provers are characterized by EXP. However, the power of $k$-prover no-signaling proofs, for $2<k<poly(n)$ remained an open problem. We show that $k$-prover no-signaling proofs (with negligible soundness) for $k=O(\sqrt{\log n})$ are contained in PSPACE. We prove this via two …
This paper studies a generalization of multi-prover interactive proofs in which a verifier interacts with two competing teams of provers: one team attempts to convince the verifier to accept while the other attempts to convince the verifier to reject. Each team consists of two provers who jointly implement a no-signaling strategy. No-signaling strategies are a curious class of joint strategy that cannot in general be implemented without communication between the provers, yet cannot be used as a black box to establish communication between them. Attention is restricted in this paper to two-turn…
We consider one-round games between a classical verifier and two provers. One of the main questions in this area is the \emph{parallel repetition question}: If the game is played $\ell$ times in parallel, does the maximum winning probability decay exponentially in $\ell$? In the classical setting, this question was answered in the affirmative by Raz. More recently the question arose whether the decay is of the form $(1-Θ(\eps))^\ell$ where $1-\eps$ is the value of the game and $\ell$ is the number of repetitions. This question is known as the \emph{strong parallel repetition question} and was …
In this paper we consider what can be computed by a user interacting with a potentially malicious server, when the server performs polynomial-time quantum computation but the user can only perform polynomial-time classical (i.e., non-quantum) computation. Understanding the computational power of this model, which corresponds to polynomial-time quantum computation that can be efficiently verified classically, is a well-known open problem in quantum computing. Our result shows that computing the order of a solvable group, which is one of the most general problems for which quantum computing exhi…
We study three variants of multi-prover quantum Merlin-Arthur proof systems. We first show that the class of problems that can be efficiently verified using polynomially many quantum proofs, each of logarithmic-size, is exactly MQA (also known as QCMA), the class of problems which can be efficiently verified via a classical proof and a quantum verifier. We then study the class BellQMA(poly), characterized by a verifier who first applies unentangled, nonadaptive measurements to each of the polynomially many proofs, followed by an arbitrary but efficient quantum verification circuit on the resul…
A central problem in quantum computational complexity is how to prevent entanglement-assisted cheating in multi-prover interactive proof systems. It is well-known that the standard oracularization technique completely fails in some proof systems under the existence of prior entanglement. This paper studies two constructions of two-prover one-round interactive proof systems based on oracularization. First, it is proved that the two-prover one-round interactive proof system for PSPACE by Cai, Condon, and Lipton still achieves exponentially small soundness error in the existence of prior entangle…
This paper gives the first formal treatment of a quantum analogue of multi-prover interactive proof systems. It is proved that the class of languages having quantum multi-prover interactive proof systems is necessarily contained in NEXP, under the assumption that provers are allowed to share at most polynomially many prior-entangled qubits. This implies that, in particular, if provers do not share any prior entanglement with each other, the class of languages having quantum multi-prover interactive proof systems is equal to NEXP. Related to these, it is shown that, in the case a prover does no…
We present a protocol that transforms any quantum multi-prover interactive proof into a nonlocal game in which questions consist of logarithmic number of bits and answers of constant number of bits. As a corollary, this proves that the promise problem corresponding to the approximation of the nonlocal value to inverse polynomial accuracy is complete for QMIP*, and therefore NEXP-hard. This establishes that nonlocal games are provably harder than classical games without any complexity theory assumptions. Our result also indicates that gap amplification for nonlocal games may be impossible in ge…
This paper considers three variants of quantum interactive proof systems in which short (meaning logarithmic-length) messages are exchanged between the prover and verifier. The first variant is one in which the verifier sends a short message to the prover, and the prover responds with an ordinary, or polynomial-length, message; the second variant is one in which any number of messages can be exchanged, but where the combined length of all the messages is logarithmic; and the third variant is one in which the verifier sends polynomially many random bits to the prover, who responds with a short …
We present upper and lower bounds of the computational complexity of the two-way communication model of multiple-prover quantum interactive proof systems whose verifiers are limited to measure-many two-way quantum finite automata. We prove that (i) the languages recognized by those multiple-prover systems running in expected polynomial time are exactly the ones in NEXP, the nondeterministic exponential-time complexity class, (ii) if we further require verifiers to be one-way quantum automata, then their associated proof systems recognize context-free languages but not beyond languages in NE, t…
The quantum PCP (QPCP) conjecture states that all problems in QMA, the quantum analogue of NP, admit quantum verifiers that only act on a constant number of qubits of a polynomial size quantum proof and have a constant gap between completeness and soundness. Despite an impressive body of work trying to prove or disprove the quantum PCP conjecture, it still remains widely open. The above-mentioned proof verification statement has also been shown equivalent to the QMA-completeness of the Local Hamiltonian problem with constant relative gap. Nevertheless, unlike in the classical case, no equivale…
Whereas quantum complexity theory has traditionally been concerned with problems arising from classical complexity theory (such as computing boolean functions), it also makes sense to study the complexity of inherently quantum operations such as constructing quantum states or performing unitary transformations. With this motivation, we define models of interactive proofs for synthesizing quantum states and unitaries, where a polynomial-time quantum verifier interacts with an untrusted quantum prover, and a verifier who accepts also outputs an approximation of the target state (for the state sy…
We introduce pseudo-deterministic interactive proofs (psdAM): interactive proof systems for search problems where the verifier is guaranteed with high probability to output the same output on different executions. As in the case with classical interactive proofs, the verifier is a probabilistic polynomial time algorithm interacting with an untrusted powerful prover. We view pseudo-deterministic interactive proofs as an extension of the study of pseudo-deterministic randomized polynomial time algorithms: the goal of the latter is to find canonical solutions to search problems whereas the goal o…
This paper proves one of the open problem posed by Beigi et al. in arXiv:1004.0411v2. We consider quantum interactive proof systems where in the beginning the verifier and prover send messages to each other with the combined length of all messages being at most logarithmic (in the input length); and at the end the prover sends a polynomial-length message to the verifier. We show that this class has the same expressive power as QMA.
We examine the power of silent constant-space probabilistic verifiers that watch asymmetric debates (where one side is unable to see some of the messages of the other) between two deterministic provers, and try to determine who is right. We prove that probabilistic verifiers outperform their deterministic counterparts as asymmetric debate checkers. It is shown that the membership problem for every language in NSPACE(s(n)) has a 2^{s(n)}-time debate where one prover is completely blind to the other one, for polynomially bounded space constructible s(n). When partial information is allowed to be…
In this paper we consider quantum interactive proof systems, i.e., interactive proof systems in which the prover and verifier may perform quantum computations and exchange quantum messages. It is proved that every language in PSPACE has a quantum interactive proof system that requires only two rounds of communication between the prover and verifier, while having exponentially small (one-sided) probability of error. It follows that quantum interactive proof systems are strictly more powerful than classical interactive proof systems in the constant-round case unless the polynomial time hierarchy…
Complexity theory traditionally studies the hardness of solving classical computational problems. In the quantum setting, it is also natural to consider a different notion of complexity, namely the complexity of physically preparing a certain quantum state. We study the relation between two such state complexity classes: statePSPACE, which contains states that can be generated by space-uniform polynomial-space quantum circuits, and stateQIP, which contains states that a polynomial-time quantum verifier can generate by interacting with an all-powerful untrusted quantum prover. The latter class …
The problem of reliably certifying the outcome of a computation performed by a quantum device is rapidly gaining relevance. We present two protocols for a classical verifier to verifiably delegate a quantum computation to two non-communicating but entangled quantum provers. Our protocols have near-optimal complexity in terms of the total resources employed by the verifier and the honest provers, with the total number of operations of each party, including the number of entangled pairs of qubits required of the honest provers, scaling as $O(g\log g)$ for delegating a circuit of size $g$. This i…
This paper studies multiple-proof quantum Merlin-Arthur (QMA) proof systems in the setting when the completeness-soundness gap is small. Small means that we only lower-bound the gap with an inverse-exponential function of the input length, or with an even smaller function. Using the protocol of Blier and Tapp [arXiv:0709.0738], we show that in this case the proof system has the same expressive power as non-deterministic exponential time (NEXP). Since single-proof QMA proof systems, with the same bound on the gap, have expressive power at most exponential time (EXP), we get a separation between…
We prove a version of the quantum de Finetti theorem: permutation-invariant quantum states are well approximated as a probabilistic mixture of multi-fold product states. The approximation is measured by distinguishability under fully one-way LOCC (local operations and classical communication) measurements. Our result strengthens Brandão and Harrow's de Finetti theorem where a kind of partially one-way LOCC measurements was used for measuring the approximation, with essentially the same error bound. As main applications, we show (i) a quasipolynomial-time algorithm which detects multipartite en…
In [20] Goldwasser, Grossman and Holden introduced pseudo-deterministic interactive proofs for search problems where a powerful prover can convince a probabilistic polynomial time verifier that a solution to a search problem is canonical. They studied search problems for which polynomial time algorithms are not known and for which many solutions are possible. They showed that whereas there exists a constant round pseudo deterministic proof for graph isomorphism where the canonical solution is the lexicographically smallest isomorphism, the existence of pseudo-deterministic interactive proofs f…
$ \newcommand{\Xlin}{\mathcal{X}} \newcommand{\Zlin}{\mathcal{Z}} \newcommand{\C}{\mathbb{C}} $We give a quantum multiprover interactive proof system for the local Hamiltonian problem in which there is a constant number of provers, questions are classical of length polynomial in the number of qubits, and answers are of constant length. The main novelty of our protocol is that the gap between completeness and soundness is directly proportional to the promise gap on the (normalized) ground state energy of the Hamiltonian. This result can be interpreted as a concrete step towards a quantum PCP th…
We present the first constructions of single-prover proof systems that achieve perfect zero knowledge (PZK) for languages beyond NP, under no intractability assumptions: 1. The complexity class #P has PZK proofs in the model of Interactive PCPs (IPCPs) [KR08], where the verifier first receives from the prover a PCP and then engages with the prover in an Interactive Proof (IP). 2. The complexity class NEXP has PZK proofs in the model of Interactive Oracle Proofs (IOPs) [BCS16,RRR16], where the verifier, in every round of interaction, receives a PCP from the prover. Our constructions rely on suc…
Many seminal results in Interactive Proofs (IPs) use algebraic techniques based on low-degree polynomials, the study of which is pervasive in theoretical computer science. Unfortunately, known methods for endowing such proofs with zero knowledge guarantees do not retain this rich algebraic structure. In this work, we develop algebraic techniques for obtaining zero knowledge variants of proof protocols in a way that leverages and preserves their algebraic structure. Our constructions achieve unconditional (perfect) zero knowledge in the Interactive Probabilistically Checkable Proof (IPCP) model…
We prove that for every 3-player (3-prover) game $\mathcal G$ with value less than one, whose query distribution has the support $\mathcal S = \{(1,0,0), (0,1,0), (0,0,1)\}$ of hamming weight one vectors, the value of the $n$-fold parallel repetition $\mathcal G^{\otimes n}$ decays polynomially fast to zero; that is, there is a constant $c = c(\mathcal G)>0$ such that the value of the game $\mathcal G^{\otimes n}$ is at most $n^{-c}$. Following the recent work of Girish, Holmgren, Mittal, Raz and Zhan (STOC 2022), our result is the missing piece that implies a similar bound for a much more gen…
Persuasion studies how an informed principal may influence the behavior of agents by the strategic provision of payoff-relevant information. We focus on the fundamental multi-receiver model by Arieli and Babichenko (2019), in which there are no inter-agent externalities. Unlike prior works on this problem, we study the public persuasion problem in the general setting with: (i) arbitrary state spaces; (ii) arbitrary action spaces; (iii) arbitrary sender's utility functions. We fully characterize the computational complexity of computing a bi-criteria approximation of an optimal public signaling…
