  We provide a unified framework for nonsignalling quantum and classical
multipartite correlations, allowing all to be written as the trace of some
local (quantum) measurements multiplied by an operator. The properties of this
operator define the corresponding set of correlations.We then show that if the
theory is such that all local quantum measurements are possible, one obtains
the correlations corresponding to the extension of Gleason's Theorem to
multipartite systems. Such correlations coincide with the quantum ones for one
and two parties, but we prove the existence of a gap for three or more parties.
