Entanglement of approximate quantum strategies in XOR games

We show that for any $\epsilon >0$ there is an XOR game $G=G(\epsilon )$ with $\mathrm{\Theta }({\epsilon }^{-1\mathrm{/}5})$ inputs for one player and $\mathrm{\Theta }({\epsilon }^{-2\mathrm{/}5})$ inputs for the other player such that $\mathrm{\Omega }({\epsilon }^{-1\mathrm{/}5})$ ebits are required for any strategy achieving bias that is at least a multiplicative factor $(1-\epsilon )$ 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 $-1\mathrm{/}5$ the scaling of our bound with $\epsilon$ is tight: for any XOR game there is an $\epsilon$-optimal strategy using $\lceil {\epsilon }^{-1}\rceil$ ebits, irrespective of the number of questions in the game.