  In classical complexity theory, the two definitions of probabilistically
checkable proofs -- the constraint satisfaction and the nonlocal games version
-- are computationally equal in power. In the quantum setting, the situation is
far less clear. The result MIP* = RE of Ji et. al. (arXiv:2001.04383) and
refinements by Natarajan and Zhang (arXiv:2302.04322) show that multiprover
interactive proof systems with polylogarithmically long messages can solve any
decision problem in RE, including undecidable problems like the halting
problem. These results show that any connection between the "constraint
satisfaction" or "Hamiltonian" quantum PCP conjecture and nonlocal games must
involve restricting the players in the game to be computationally efficient.
This note contains two main results: (1) we give a "quantum games PCP for AM"
in the form of a new construction of a succinct MIP* protocol with efficient
provers for the canonical AM-complete problem, and (2) we explain an error in
the energy amplification procedure of Natarajan and Vidick (arXiv:1710.03062)
which invalidates their claim to have constructed a quantum games PCP for a
QMA-complete problem. In surveying the obstacles remaining towards a quantum
games PCP for QMA, we highlight the importance and challenge of understanding
gap amplification for Hamiltonians even when locality is replaced by much
weaker constraints, such as bounds on the "Pauli spectrum" of the Hamiltonian.
We hope these questions will motivate progress towards new "baby versions" of
Hamiltonian quantum PCP conjecture.
