Entanglement of approximate quantum strategies in XOR games

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