  Nonlocal games play a crucial role in quantum information theory and have
numerous applications in certification and cryptographic protocols. Kalai et
al. (STOC 2023) introduced a procedure to compile a nonlocal game into a
single-prover interactive proof, using a quantum homomorphic encryption scheme,
and showed that their compilation method preserves the classical bound of the
game. Natarajan and Zhang (FOCS 2023) then showed that the quantum bound is
preserved for the specific case of the CHSH game. Extending the proof
techniques of Natarajan and Zhang, we show that the compilation procedure of
Kalai et al. preserves the quantum bound for two classes of games: XOR games
and d-outcome CHSH games. We also establish that, for any pair of qubit
measurements, there exists an XOR game such that its optimal winning
probability serves as a self-test for that particular pair of measurements.
