In a $3$-$\mathsf{XOR}$ game $\mathcal{G}$, the verifier samples a challenge $(x,y,z)\sim μ$ where $μ$ is a probability distribution over $Σ\timesΓ\timesΦ$, and a map $t\colon Σ\timesΓ\timesΦ\to\mathcal{A}$ for a finite Abelian group $\mathcal{A}$ defining a constraint. The verifier sends the questions $x$, $y$ and $z$ to the players Alice, Bob and Charlie respectively, receives answers $f(x)$, $g(y)$ and $h(z)$ that are elements in $\mathcal{A}$ and accepts if $f(x)+g(y)+h(z) = t(x,y,z)$. The value, $\mathsf{val}(\mathcal{G})$, of the game is defined to be the maximum probability the verifier accepts over all players' strategies.
We show that if $\mathcal{G}$ is a $3$-$\mathsf{XOR}$ game with value strictly less than $1$, whose underlying distribution over questions $μ$ does not admit Abelian embeddings into $(\mathbb{Z},+)$, then the value of the $n$-fold repetition of $\mathcal{G}$ is exponentially decaying. That is, there exists $c=c(\mathcal{G})>0$ such that $\mathsf{val}(\mathcal{G}^{\otimes n})\leq 2^{-cn}$. This extends a previous result of [Braverman-Khot-Minzer, FOCS 2023] showing exponential decay for the GHZ game. Our proof combines tools from additive combinatorics and tools from discrete Fourier analysis.
We show that for any eps>0 the problem of finding a factor (2-eps) approximation to the entangled value of a three-player XOR game is NP-hard. Equivalently, the problem of approximating the largest possible quantum violation of a tripartite Bell correlation inequality to within any multiplicative constant is NP-hard. These results are the first constant-factor hardness of approximation results for entangled games or quantum violations of Bell inequalities shown under the sole assumption that P \neq NP. They can be thought of as an extension of Hastad's optimal hardness of approximation results…
We prove that for every 3-player game with binary questions and answers and value $<1$, the value of the $n$-fold parallel repetition of the game decays polynomially fast to 0. That is, for every such game, there exists a constant $c>0$, such that the value of the $n$-fold parallel repetition of the game is at most $n^{-c}$. Along the way to proving this theorem, we prove two additional parallel repetition theorems for multiplayer games, that may be of independent interest: Playerwise Connected Games (with any number of players and any Alphabet size): We identify a large class of multiplayer g…
We give a new proof of the fact that the parallel repetition of the (3-player) GHZ game reduces the value of the game to zero polynomially quickly. That is, we show that the value of the $n$-fold GHZ game is at most $n^{-Ω(1)}$. This was first established by Holmgren and Raz [HR20]. We present a new proof of this theorem that we believe to be simpler and more direct. Unlike most previous works on parallel repetition, our proof makes no use of information theory, and relies on the use of Fourier analysis. The GHZ game [GHZ89] has played a foundational role in the understanding of quantum inform…
We study the complexity of computing the commuting-operator value $ω^*$ of entangled XOR games with any number of players. We introduce necessary and sufficient criteria for an XOR game to have $ω^* = 1$, and use these criteria to derive the following results: 1. An algorithm for symmetric games that decides in polynomial time whether $ω^* = 1$ or $ω^* < 1$, a task that was not previously known to be decidable, together with a simple tensor-product strategy that achieves value 1 in the former case. The only previous candidate algorithm for this problem was the Navascués-Pironio-Acín (also know…
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…
We prove that parallel repetition of the (3-player) GHZ game reduces the value of the game polynomially fast to 0. That is, the value of the GHZ game repeated in parallel $t$ times is at most $t^{-Ω(1)}$. Previously, only a bound of $\approx \frac{1}{α(t)}$, where $α$ is the inverse Ackermann function, was known. The GHZ game was recently identified by Dinur, Harsha, Venkat and Yuen as a multi-player game where all existing techniques for proving strong bounds on the value of the parallel repetition of the game fail. Indeed, to prove our result we use a completely new proof technique. Dinur, H…
In this paper, we prove a strong XOR lemma for bounded-round two-player randomized communication. For a function $f:\mathcal{X}\times \mathcal{Y}\rightarrow\{0,1\}$, the $n$-fold XOR function $f^{\oplus n}:\mathcal{X}^n\times \mathcal{Y}^n\rightarrow\{0,1\}$ maps $n$ input pairs $(X_1,\ldots,X_n,Y_1,\ldots,Y_n)$ to the XOR of the $n$ output bits $f(X_1,Y_1)\oplus \cdots \oplus f(X_n, Y_n)$. We prove that if every $r$-round communication protocols that computes $f$ with probability $2/3$ uses at least $C$ bits of communication, then any $r$-round protocol that computes $f^{\oplus n}$ with proba…
We prove that a sufficiently strong parallel repetition theorem for a special case of multiplayer (multiprover) games implies super-linear lower bounds for multi-tape Turing machines with advice. To the best of our knowledge, this is the first connection between parallel repetition and lower bounds for time complexity and the first major potential implication of a parallel repetition theorem with more than two players. Along the way to proving this result, we define and initiate a study of block rigidity, a weakening of Valiant's notion of rigidity. While rigidity was originally defined for ma…
We introduce a simple transformation on two-player nonlocal games, called "anchoring", and prove an exponential-decay parallel repetition theorem for all anchored games in the setting of quantum entangled players. This transformation is inspired in part by the Feige-Kilian transformation (SICOMP 2000), and has the property that if the quantum value of the original game $G$ is $v$ then the quantum value of the anchored game $G_\bot$ is $1 - (1 - α)^2 \cdot (1 - v)$ where $α$ is a parameter of the transformation. In particular the anchored game has quantum value $1$ if and only if the original g…
For distinguishing quantum states sampled from a fixed ensemble, the gap in bipartite and single-party distinguishability can be interpreted as a nonlocality of the ensemble. In this paper, we consider bipartite state discrimination in a composite system consisting of $N$ subsystems, where each subsystem is shared between two parties and the state of each subsystem is randomly sampled from a particular ensemble comprising the Bell states. We show that the success probability of perfectly identifying the state converges to $1$ as $N\rightarrow\infty$ if the entropy of the probability distributi…
We prove a direct product theorem for the one-way entanglement-assisted quantum communication complexity of a general relation $f\subseteq\mathcal{X}\times\mathcal{Y}\times\mathcal{Z}$. For any $\varepsilon, ζ> 0$ and any $k\geq1$, we show that \[ \mathrm{Q}^1_{1-(1-\varepsilon)^{Ω(ζ^6k/\log|\mathcal{Z}|)}}(f^k) = Ω\left(k\left(ζ^5\cdot\mathrm{Q}^1_{\varepsilon + 12ζ}(f) - \log\log(1/ζ)\right)\right),\] where $\mathrm{Q}^1_{\varepsilon}(f)$ represents the one-way entanglement-assisted quantum communication complexity of $f$ with worst-case error $\varepsilon$ and $f^k$ denotes $k$ parallel ins…
