  The foundation of zero-knowledge is the simulator: a weak machine capable of
pretending to be a weak verifier talking with all-powerful provers. To achieve
this, simulators need some kind of advantage such as the knowledge of a
trapdoor. In existing zero-knowledge multi-prover protocols, this advantage is
essentially signalling, something that the provers are explicitly forbidden to
do. In most cases, this advantage is stronger than necessary as it is possible
to define a sense in which simulators need much less to simulate. We define a
framework in which we can quantify the simulators' non-local advantage and
exhibit examples of zero-knowledge protocols that are sound against local or
entangled provers but that are not sound against no-signalling provers
precisely because the no-signalling simulation strategy can be adopted by
malicious provers.
