For four-player XOR games, commuting-operator value one forces a perfect GHZ-equatorial strategy exactly when each player has at most three active questions; the Klein four-group game shows the threshold is sharp.
and Wolf, Michael M
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A Sharp Local-Question Threshold for GHZ-Equatorial Completeness in Four-Player XOR Games
For four-player XOR games, commuting-operator value one forces a perfect GHZ-equatorial strategy exactly when each player has at most three active questions; the Klein four-group game shows the threshold is sharp.