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

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 $8\mathrm{/}9$ for all $N$, while for $N\ge 3$, a generalization of the quantum strategy for $N=2$ achieves a winning probability between $1\mathrm{/}3$ and $3\mathrm{/}4$. This loss of quantum advantage for $N\ge 3$ is a consequence of monogamy.