  We consider the problem of the classical simulation of quantum measurements
in the scenario of communication complexity. Regev and Toner (2007) have
presented a 2-bit protocol which simulates one particular correlation function
arising from binary projective quantum measurements on arbitrary state, and in
particular does not preserve local averages. The question of simulating other
correlation functions using a protocol with bounded communication, or
preserving local averages, has been posed as an open one. Within this paper we
resolve it in the negative: we show that any such protocol must have unbounded
communication for some subset of executions. In particular, we show that for
any protocol, there exist inputs for which the random variable describing the
number of communicated bits has arbitrarily large variance.
