We introduce quantum XOR games, a model of two-player, one-round games that extends the model of XOR games by allowing the referee’s questions to the players to be quantum states. We give examples showing that quantum XOR games exhibit a wide range of behaviors that are known not to exist for standard XOR games, such as cases in which the use of entanglement leads to an arbitrarily large advantage over the use of no entanglement. By invoking two deep extensions of Grothendieck’s inequality, we present an efficient algorithm that gives a constant-factor approximation to the best performance that players can obtain in a given game, both in the case that they have no shared entanglement and that they share unlimited entanglement. As a byproduct of the algorithm, we prove some additional interesting properties of quantum XOR games, such as the fact that sharing a maximally entangled state of arbitrary dimension gives only a small advantage over having no entanglement at all.