Randomness in nonlocal games between mistrustful players

If two quantum players at a nonlocal game G achieve a superclassical score, then their measurement outcomes must be at least partially random from the perspective of any third player. This is the basis for device-independent quantum cryptography. In this paper we address a related question: does a superclassical score at G guarantee that one player has created randomness from the perspective of the other player? We show that for complete-support games, the answer is yes: even if the second player is given the first player’s input at the conclusion of the game, he cannot perfectly recover her output. This implies that some amount of local randomness (i.e., randomness possessed by only one player) can always be obtained when randomness is certified by nonlocal games. We discuss potential implications for cryptographic protocols between mistrustful parties.