Entanglement of approximate quantum strategies in XOR games

We show that for any ε > 0 there is an XOR game G = G(ε) with Θ(ε−1/5) inputs for one player and Θ(ε−2/5) inputs for the other player such that Ω(ε−1/5) ebits are required for any strategy achieving bias that is at least a multiplicative factor (1 − ε) from optimal. This gives an exponential improvement in both the number of inputs or outputs and the noise tolerance of any previously-known self-test for highly entangled states. Up to the exponent −1/5 the scaling of our bound with ε is tight: for any XOR game there is an ε-optimal strategy using ⌈ε−1⌉ ebits, irrespective of the number of questions in the game.