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arXiv:2610.30455

Parallel Repetition for k-Player XOR Games

Mark Braverman, Dor Minzer

Published Oct 21, 2026• 0 citations
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Abstract

In a $k$-XOR game $Ψ$ over $\mathbb{F}_p$, a verifier samples challenges $(x_1,\ldots,x_k)\simμ$ where $μ$ is a distribution over $Σ_1\times\ldots\times Σ_k$, sends $x_i$ to player $i$ and receives an answer $f_i(x_i)\in\mathbb{F}_p$, and accepts if and only if $f_1(x_1)+\ldots+f_k(x_k)=t(x_1,\ldots,x_k)$, where $t\colon Σ_1\times\ldots\times Σ_k\to\mathbb{F}_p$ is a fixed map. We show that if $Ψ$ has value strictly less than $1$, then the value of the $n$-fold repeated game $Ψ^{\otimes n}$ is at most $2^{-cn}$ where $c = c(Ψ)>0$ is a fixed constant. This extends a previous result of [Bhangale-Braverman-Khot-Liu-Minzer] showing exponential decay for some $3$-XOR games. Our proof uses a relaxation of the value of games studied in the quantum setting, called the commuting value of a game. We show a transformation that maps a $k$-player game $Ψ$ with value strictly less than $1$ to a $2k$-player game $Ψ'$ with commuting value less than $1$, such that $\mathsf{val}(Ψ^{\otimes n})\leq \mathsf{val}((Ψ')^{\otimes n})$. We prove that $\mathsf{val}(Ψ')<1$ has a direct proof using nothing more than the Cauchy-Schwarz inequality, which we are able to tensorize to get that $\mathsf{val}((Ψ')^{\otimes n})$ vanishes exponentially fast in $n$.

DOI
10.48550/arxiv.2610.30455

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We study the behavior of the entangled value of two-player one-round projection games under parallel repetition. We show that for any projection game GGG of entangled value 1-eps < 1, the value of the kkk-fold repetition of G goes to zero as O((1-eps^c)^k), for some universal constant c\geq 1. Previously parallel repetition with an exponential decay in kkk was only known for the case of XOR and unique games. To prove the theorem we extend an analytical framework recently introduced by Dinur and Steurer for the study of the classical value of projection games under parallel repetition. Our proof, as theirs, relies on the introduction of a simple relaxation of the entangled value that is perfectly multiplicative. The main technical component of the proof consists in showing that the relaxed value remains tightly connected to the entangled value, thereby establishing the parallel repetition theorem. More generally, we obtain results on the behavior of the entangled value under products of arbitrary (not necessarily identical) projection games. Relating our relaxed value to the entangled value is done by giving an algorithm for converting a relaxed variant of quantum strategies that we call "vector quantum strategy" to a quantum strategy. The algorithm is considerably simpler in case the bipartite distribution of questions in the game has good expansion properties. When this is not the case, rounding relies on a quantum analogue of Holenstein's correlated sampling lemma which may be of independent interest. Our "quantum correlated sampling lemma" generalizes results of van Dam and Hayden on universal embezzlement.
We prove that for every 3-player (3-prover) game G\mathcal GG with value less than one, whose query distribution has the support S={(1,0,0),(0,1,0),(0,0,1)}\mathcal S = \{(1,0,0), (0,1,0), (0,0,1)\}S={(1,0,0),(0,1,0),(0,0,1)} of hamming weight one vectors, the value of the nnn-fold parallel repetition G⊗n\mathcal G^{\otimes n}G⊗n decays polynomially fast to zero; that is, there is a constant c=c(G)>0c = c(\mathcal G)>0c=c(G)>0 such that the value of the game G⊗n\mathcal G^{\otimes n}G⊗n is at most n−cn^{-c}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 general class of multiplayer games: For every\textbf{every}every 3-player game G\mathcal GG over binary questions\textit{binary questions}binary questions and arbitrary answer lengths\textit{arbitrary answer lengths}arbitrary answer lengths, with value less than 1, there is a constant c=c(G)>0c = c(\mathcal G)>0c=c(G)>0 such that the value of the game G⊗n\mathcal G^{\otimes n}G⊗n is at most n−cn^{-c}n−c. Our proof technique is new and requires many new ideas. For example, we make use of the Level-kkk inequalities from Boolean Fourier Analysis, which, to the best of our knowledge, have not been explored in this context prior to our work.
In a two-player game, two cooperating but non communicating players, Alice and Bob, receive inputs taken from a probability distribution. Each of them produces an output and they win the game if they satisfy some predicate on their inputs/outputs. The entangled value ω∗(G)ω^*(G)ω∗(G) of a game GGG is the maximum probability that Alice and Bob can win the game if they are allowed to share an entangled state prior to receiving their inputs. The nnn-fold parallel repetition GnG^nGn of GGG consists of nnn instances of GGG where Alice and Bob receive all the inputs at the same time and must produce all the outputs at the same time. They win GnG^nGn if they win each instance of GGG. Recently, there has been a series of works showing parallel repetition with exponential decay for projection games [DSV13], games on the uniform distribution [CS14] and for free games, i.e. games on a product distribution [JPY13]. This article is meant to be a follow up of [CS14], where we improve and simplify several parts of our previous paper. Our main result is that for any free game GGG with value ω∗(G)=1−εω^*(G)=1-\varepsilonω∗(G)=1−ε, we have ω∗(Gn)≤(1−ε2)Ω(nlog⁡(l))ω^*(G^n) \le (1 - \varepsilon^2)^{Ω(\frac{n}{\log(l)})}ω∗(Gn)≤(1−ε2)Ω(log(l)n​) where lll is the size of the output set of the game. This result improves on both the results in [JPY13] and [CS14]. The framework we use can also be extended to free projection games. We show that for a free projection game GGG with value ω∗(G)=1−εω^*(G)=1-\varepsilonω∗(G)=1−ε, we have ω∗(Gn)≤(1−ε)Ω(n)ω^*(G^n) \le (1 - \varepsilon)^{Ω(n)}ω∗(Gn)≤(1−ε)Ω(n).
The behavior of games repeated in parallel, when played with quantumly entangled players, has received much attention in recent years. Quantum analogues of Raz's classical parallel repetition theorem have been proved for many special classes of games. However, for general entangled games no parallel repetition theorem was known. We prove that the entangled value of a two-player game G repeated n times in parallel is at most c_G*n^{-1/4}*log(n) for a constant c_G depending on G, provided that the entangled value of G is less than 1. In particular, this gives the first proof that the entangled value of a parallel repeated game must converge to 0 for all games whose entangled value is less than 1. Central to our proof is a combination of both classical and quantum correlated sampling.
We study the behavior of the entangled value of two-player one-round projection games under parallel repetition. We show that for any projection game G of entangled value 1 - εc)k), for some universal constant c ≥ 1. Previously parallel repetition with an exponential decay in k was only known for the case of XOR and unique games. To prove the theorem we extend an analytical framework recently introduced by Dinur and Steurer for the study of the classical value of projection games under parallel repetition. Our proof, as theirs, relies on the introduction of a simple relaxation of the entangled value that is perfectly multiplicative. The main technical component of the proof consists in showing that the relaxed value remains tightly connected to the entangled value, thereby establishing the parallel repetition theorem. More generally, we obtain results on the behavior of the entangled value under products of arbitrary (not necessarily identical) projection games. Relating our relaxed value to the entangled value is done by giving an algorithm for converting a relaxed variant of quantum strategies that we call “vector quantum strategy” to a quantum strategy. The algorithm is considerably simpler in case the bipartite distribution of questions in the game has good expansion properties. When this is not the case, rounding relies on a quantum analogue of Holenstein's correlated sampling lemma which may be of independent interest. Our “quantum correlated sampling lemma” generalizes results of van Dam and Hayden on universal embezzlement to the following approximate scenario: two isolated parties, given classical descriptions of arbitrary bipartite states |ψ〉, |φ〉 respectively such that |ψ〉 ≈ |φ〉, are able to locally generate a joint entangled state|Ψ〉 ≈ |ψ〉 ≈ |φ〉 using an initial entangled state that is independent of their inputs.