  We present a step towards the goal of producing a general cryptographic
'compilation' procedure which can translate any entangled nonlocal game into a
single-prover interactive protocol while preserving quantum completeness and
soundness, using cryptography to simulate the separation between the provers. A
candidate for such a procedure was introduced by Kalai et al. (STOC '23), who
defined a black-box cryptographic compilation procedure that applies to any
nonlocal game and showed that it preserves classical value. In this work, we
make progress towards a full understanding of the quantum value of the
single-prover protocols that result from applying the Kalai et al. compilation
procedure to entangled games.
  For the special case of CHSH, we prove that the Tsirelson bound holds under
the compilation procedure introduced by Kalai et al., and we also recover a
strong version of the 'rigidity' property that makes CHSH so useful. As an
application, we give a single-prover cryptographically sound classical
verification protocol for BQP, and we prove its soundness using our CHSH
rigidity analysis. Our protocol replicates the functionality of Mahadev's
protocol (FOCS '18) but with two advantages: (1) the protocol is conceptually
intuitive and requires fewer bespoke ingredients, and the soundness analysis is
simpler and directly follows the analysis of the nonlocal case, and (2) the
soundness analysis does not explicitly use the assumption of a TCF or an
adaptive hardcore bit, and only requires QFHE as a black box (though currently
the only known constructions of QFHE use TCFs).
