  A central problem in quantum computational complexity is how to prevent
entanglement-assisted cheating in multi-prover interactive proof systems. It is
well-known that the standard oracularization technique completely fails in some
proof systems under the existence of prior entanglement. This paper studies two
constructions of two-prover one-round interactive proof systems based on
oracularization. First, it is proved that the two-prover one-round interactive
proof system for PSPACE by Cai, Condon, and Lipton still achieves exponentially
small soundness error in the existence of prior entanglement between dishonest
provers (and more strongly, even if dishonest provers are allowed to use
arbitrary no-signaling strategies). It follows that, unless the polynomial-time
hierarchy collapses to the second level, two-prover systems are still
advantageous to single-prover systems even when only malicious provers can use
quantum information. Second, it is proved that the two-prover one-round
interactive proof system obtained by oracularizing a three-query
probabilistically checkable proof system becomes sound in a weak sense even
against dishonest entangled provers with the help of a dummy question. As a
consequence, every language in NEXP has a two-prover one-round interactive
proof system of perfect completeness, albeit with exponentially small gap
between completeness and soundness, in which each prover responds with only two
bits. In other words, it is NP-hard to approximate within an inverse-polynomial
the value of a classical two-prover one-round game, even when provers are
entangled and each sends a two-bit answer to a verifier.
