There exist pairs of orthogonal Latin squares of any order $n$ except if $n=2$ or $n=6$ [Bose et al., Further results on the construction of mutually orthogonal latin squares and the falsity of Euler’s conjecture, Can. J. Math. 12, 189 (1960)]. In particular, the problem of Euler’s thirty-six officers does not have a solution. However, it has a “quantum solution”: there exist so-called entangled quantum Latin squares of order 6 [Rather et al., Thirty-six entangled officers of Euler: Quantum solution to a classically impossible problem, Phys. Rev. Lett. 128, 080507 (2022)]. We prove that mutually orthogonal quantum Latin squares of order 6 do not exist if entanglement is not allowed.

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• Quantum entanglement