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arXiv:2610.06534

The Mermin-Peres Magic Square Game With Three or More Players

Itamar Lorian, J. E. Avron

quant-phSubmitted Oct 5, 2026• Announced Oct 6, 2026• 0 citations• Revised Oct 14, 2026 (v2)
v1 · Oct 5, 2026v2 · Oct 14, 2026
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Abstract

We present a generalization of the Mermin-Peres magic square game to $N$ players, in which the square is replaced by an $N$-dimensional hypercube. We show that the winning probability of the optimal classical strategy remains $8/9$ for all $N$, while for $N \ge 3$, a generalization of the quantum strategy for $N = 2$ achieves a winning probability between $1/3$ and $3/4$. This loss of quantum advantage for $N \ge 3$ is a consequence of monogamy.

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12 paged, 4 figures

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We present a generalization of the Mermin-Peres magic square game to NNN players, in which the square is replaced by an NNN-dimensional hypercube. We show that the winning probability of the optimal classical strategy remains 8/98/98/9 for all NNN, while for N≥3N \ge 3N≥3, a generalization of the quantum strategy for N=2N = 2N=2 achieves a winning probability between 1/31/31/3 and 3/43/43/4. This loss of quantum advantage for N≥3N \ge 3N≥3 is a consequence of monogamy.
The Mermin-Peres magic square game is a cooperative two-player nonlocal game in which shared quantum entanglement allows the players to win with certainty, while players limited to classical operations cannot do so, a phenomenon dubbed "quantum pseudo-telepathy". The game has a referee separately ask each player to color a subset of a 3x3 grid. The referee checks that their colorings satisfy certain parity constraints that can't all be simultaneously realized. We define a generalization of these games to be played on an arbitrary arrangement of intersecting sets of elements. We characterize exactly which of these games exhibit quantum pseudo-telepathy, and give quantum winning strategies for those that do. In doing so, we show that it suffices for the players to share three Bell pairs of entanglement even for games on arbitrarily large arrangements. Moreover, it suffices for Alice and Bob to use measurements from the three-qubit Pauli group. The proof technique uses a novel connection of Mermin-style games to graph planarity.
Prior to this work, the largest known gap between the entangled value and the classical value for a one-round two-player nonlocal game with a perfect entangled strategy using two Bell states of entanglement was 19\frac{1}{9}91​, achieved by the Mermin-Peres magic square game. A larger gap of 215\frac{2}{15}152​ has been claimed for the related doily game of Kelleher, Roomy, and Holweck (Quant. Inf. Proc., 2024). However, this value assumes that the classical players use identical local strategies, and we show that dropping this assumption reduces the gap back to 19\frac{1}{9}91​. We then demonstrate that 19\frac{1}{9}91​ is not optimal by explicitly constructing a new nonlocal game with gap 435\frac{4}{35}354​.
We show that the nnn-round parallel repetition of the Magic Square game of Mermin and Peres is rigid, in the sense that for any entangled strategy succeeding with probability 1−ε1 -\varepsilon1−ε, the players' shared state is O(poly(nε))O(\mathrm{poly}(n\varepsilon))O(poly(nε))-close to 2n2n2n EPR pairs under a local isometry. Furthermore, we show that, under local isometry, the players' measurements in said entangled strategy must be O(poly(nε))O(\mathrm{poly}(n\varepsilon))O(poly(nε)) close to the "ideal" strategy when acting on the shared state.
We introduce a novel generalization of the Clauser-Horne-Shimony-Holt (CHSH) game to a multiplayer setting, i.e., Hypercube game, where all mmm players are required to assign values to vertices on corresponding facets of an mmm-dimensional hypercube. The players win if and only if their answers satisfy both parity and consistency conditions. We completely characterize the maximum winning probabilities (game value) under classical, quantum and no-signalling strategies, respectively. In contrast to the original CHSH game designed to demonstrate the superiority of quantumness, we find that the quantum advantages in the Hypercube game significantly decrease as the number of players increase. Notably, the quantum value decays exponentially fast to the classical value as mmm increases, while the no-signalling value always remains to be one.
Quantum games embody non-intuitive consequences of quantum phenomena, such as entanglement and contextuality. The Mermin-Peres game is a simple example, demonstrating how two players can utilise shared quantum information to win a no - communication game with certainty, where classical players cannot. In this paper we look at the geometric structure behind such classical strategies, and borrow ideas from the geometry of symplectic polar spaces to maximise this quantum advantage. We introduce a new game called the Eloily game with a quantum-classical success gap of 0.26‾0.2\overline{6}0.26, larger than that of the Mermin-Peres and doily games. We simulate this game in the IBM Quantum Experience and obtain a success rate of 111, beating the classical bound of 0.73‾0.7\overline{3}0.73 demonstrating the efficiency of the quantum strategy.
Abstract Quantum games embody non-intuitive consequences of quantum phenomena, such as entanglement and contextuality. The Mermin-Peres game is a simple example, demonstrating how two players can utilise shared quantum information to win a no—communication game with certainty, where classical players cannot. In this paper we look at the geometric structure behind such classical strategies, and borrow ideas from the geometry of symplectic polar spaces to maximise this quantum advantage. We introduce a new game called the Eloily game with a quantum–classical success gap of 0.2 6 ― , larger than that of the Mermin-Peres and doily games. We simulate this game in the IBM Quantum Experience and obtain a success rate of 1, beating the classical bound of 0.7 3 ― demonstrating the efficiency of the quantum strategy.