Classical, Quantum and Non-signalling Resources in Bipartite Games

We study bipartite games that arise in the context of nonlocality with the help of graph theory. Our main results are alternate proofs that deciding whether a no-communication classical winning strategy exists for certain games (called forbidden-edge and covering games) is NP- complete, while the problem of deciding if these games admit a non-signalling winning strategy is in P. We discuss relations between quantum winning strategies and orthogonality graphs. We also show that every pseudo-telepathy game yields both a proof of the Bell-Kochen-Specker theorem and an instance of a two-prover interactive proof system that is classically sound, but that becomes unsound when provers use shared entanglement.