  Communication complexity problems (CCPs) are tasks in which separated parties
attempt to compute a function whose inputs are distributed among the parties.
Their communication is limited so that not all inputs can be sent. We show that
broad classes of Bell inequalities can be mapped to CCPs and that a quantum
violation of a Bell inequality is a necessary and sufficient condition for an
enhancement of the related CCP beyond its classical limitation. However, one
can implement CCPs by transmitting a quantum system, encoding no more
information than is allowed in the CCP, and extract information by performing
measurements. We show that for a large class of Bell inequalities, the
improvement of the CCP associated to a quantum violation of a Bell inequality
can be no greater than the improvement obtained from quantum
prepare-transmit-measure strategies.
