  We consider two-stage hybrid protocols that combine quantum resource and
classical resource to generate classical correlations shared by two separated
players. Our motivation is twofold. First, in the near future the scale of
quantum information processing is quite limited, and when quantum resource
available is not sufficient for certain tasks, a possible way to strengthen the
capability of quantum schemes is introducing extra classical resource. We
analyze the mathematical structures of these hybrid protocols, and characterize
the relation between the amount of quantum resource and classical resource
needed. Second, a fundamental open problem in communication complexity theory
is to describe the advantages of sharing prior quantum entanglement over
sharing prior randomness, which is still widely open. It turns out that our
quantum and classical hybrid protocols provide new insight into this important
problem.
