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arXiv:2607.06876

XOR Games at Full Tilt: The Hardness of Binary Nonlocal Games

Richard Cleve, Eric Culf, Aviv Taller

quant-phcs.CCSubmitted Jul 8, 2026• Announced Jul 9, 2026• 0 citations
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Abstract

It is well known that the quantum value of an XOR nonlocal game, where the winning condition depends only on the XOR of the two players' output bits, may be approximated in polynomial time. We study a variant of the XOR game model, which we call tilted XOR games, where the winning condition can additionally depend on only one of the output bits. We show that this dramatically increases the expressive power: the computational complexity of the problem of approximating the quantum value of tilted XOR games to constant precision is RE-complete. Also, our result extends to succinct versions of tilted XOR games, where the questions can be polynomial-length binary strings, generated by a polynomial-time verifier. For classical strategies, the distinction between XOR games and tilted XOR games is inconsequential. Håstad (J. ACM, 2001) shows that they are both NP-complete to approximate, by using a reduction from linear systems to XOR games. Our approach is to show that this is also quantum-sound, but as a reduction from linear system games to tilted XOR games. Since titled XOR games are a special case of binary games (where each party outputs a single bit), our result implies that binary games are RE-hard to approximate.

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44 pages, 4 figures

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It is well known that the quantum value of an XOR nonlocal game, where the winning condition depends only on the XOR of the two players' output bits, may be approximated in polynomial time. We study a variant of the XOR game model, which we call tilted XOR games, where the winning condition can additionally depend on only one of the output bits. We show that this dramatically increases the expressive power: the computational complexity of the problem of approximating the quantum value of tilted XOR games to constant precision is RE-complete. Also, our result extends to succinct versions of tilted XOR games, where the questions can be polynomial-length binary strings, generated by a polynomial-time verifier.
For classical strategies, the distinction between XOR games and tilted XOR games is inconsequential. Håstad (J. ACM, 2001) shows that they are both NP-complete to approximate, by using a reduction from linear systems to XOR games. Our approach is to show that this is also quantum-sound, but as a reduction from linear system games to tilted XOR games.
Since titled XOR games are a special case of binary games (where each party outputs a single bit), our result implies that binary games are RE-hard to approximate.
We show that for any ε>0\varepsilon>0ε>0 the problem of finding a factor (2−ε)(2-\varepsilon)(2−ε) 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 H\aastad's optimal hardness of approximation results for MAX-E3-LIN2 [J. ACM, 48 (2001), pp. 798--859] to the entangled-player setting. The key technical component of our work is a soundness analysis of a plane-vs-point low-degree test against entangled players. This extends and simplifies the analysis of the multilinearity test by Ito and Vidick [Proceedings of the 535353rd FOCS, IEEE, Piscataway, NJ, 2012, pp. 243--252]. Our results demonstrate the possibility of efficient reductions between entangled-player games and our techniques may lead to further hardness of approximation results.
Two-party xor games (correlation Bell inequalities with two outcomes per party) are the most studied Bell inequalities, and one of the few classes for which the optimal quantum value is known to be exactly calculable. We study a natural generalization of the binary xor games to the class of linear games with d>2d>2d>2 outcomes, and propose an easily computable bound on the quantum value of these games. Many interesting properties such as the impossibility of a quantum strategy to win these games, and the quantum bound on the CHSH game generalized to ddd outcomes are derived. We also use the proposed bound to prove a large-alphabet generalization of the principle of no quantum advantage in nonlocal computation, showing that quantum theory provides no advantage in the task of nonlocal distributed computation of a class of functions with ddd outcomes for prime ddd, while general no-signaling boxes do. This task is one of the information-theoretic principles attempting to characterize the set of quantum correlations from amongst general no-signaling ones.
A two-player one-round binary game consists of two cooperative players who each replies by one bit to a message that he receives privately; they win the game if both questions and answers satisfy some predetermined property. A game is called entangled if the players are allowed to share a priori entanglement. It is well-known that the maximum winning probability (value) of entangled XOR-games (binary games in which the predetermined property depends only on the XOR of the two output bits) can be computed by a semidefinite program. In this paper we extend this result in the following sense; if a binary game is uniform, meaning that in an optimal strategy the marginal distributions of the output of each player are uniform, then its entangled value can be efficiently computed by a semidefinite program. We also introduce a lower bound on the entangled value of a general two-player one-round game; this bound depends on the size of the output set of each player and can be computed by a semidefinite program. In particular, we show that if the game is binary, ωq\omega_qωq​ is its entangled value, and ωsdp\omega_{sdp}ωsdp​ is the optimum value of the corresponding semidefinite program, then 0.68 ωsdp<ωq≤ωsdp0.68\,\omega_{sdp} < \omega_q \leq \omega_{sdp}0.68ωsdp​<ωq​≤ωsdp​.
A two-player one-round binary game consists of two cooperative players who each replies by one bit to a message that he receives privately; they win the game if both questions and answers satisfy some predetermined property. A game is called entangled if the players are allowed to share a priori entanglement. It is well-known that the maximum winning probability (value) of entangled XOR-games (binary games in which the predetermined property depends only on the XOR of the two output bits) can be computed by a semidefinite program. In this paper we extend this result in the following sense; if a binary game is uniform, meaning that in an optimal strategy the marginal distributions of the output of each player are uniform, then its entangled value can be efficiently computed by a semidefinite program. We also introduce a lower bound on the entangled value of a general two-player one-round game; this bound depends on the size of the output set of each player and can be computed by a semidefinite program. In particular, we show that if the game is binary, w_q is its entangled value, and w_{sdp} is the optimum value of the corresponding semidefinite program, then 0.68w_{sdp} < w_q <= w_{sdp}.
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 for MAX-E3-LIN2 (JACM'01) to the entangled-player setting. The key technical component of our work is a soundness analysis of a point-vs-plane low-degree test against entangled players. This extends and simplifies the analysis of the multilinearity test by Ito and Vidick (FOCS'12). Our results demonstrate the possibility for efficient reductions between entangled-player games and our techniques may lead to further hardness of approximation results.