We show that for any $arepsilon>0$ there is an XOR game $G=G(arepsilon)$with $Theta(arepsilon^{-1/5})$ inputs for one player and$Theta(arepsilon^{-2/5})$ inputs for the other player such that$Omega(arepsilon^{-1/5})$ ebits are required for any strategy achieving biasthat is at least a multiplicative factor $(1-arepsilon)$ from optimal. Thisgives an exponential improvement in both the number of inputs or outputs andthe noise tolerance of any previously-known self-test for highly entangledstates. Up to the exponent $-1/5$ the scaling of our bound with $arepsilon$is tight: for any XOR game there is an $arepsilon$-optimal strategy using$lceil arepsilon^{-1} ceil$ ebits, irrespective of the number of questionsin the game.
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