  Quantum theory is known to be nonlocal in the sense that separated parties
can perform measurements on a shared quantum state to obtain correlated
probability distributions, which cannot be achieved if the parties share only
classical randomness. Here we find that the set of distributions compatible
with sharing quantum states subject to some sufficiently restricted dimension
is neither convex nor a superset of the classical distributions. We examine the
relationship between quantum distributions associated with a dimensional
constraint and classical distributions associated with limited shared
randomness. We prove that quantum correlations are convex for certain finite
dimension in certain Bell scenarios and that they sometimes offer a dimensional
advantage in realizing local distributions. We also consider if there exist
Bell scenarios where the set of quantum correlations is never convex with
finite dimensionality.
