  Communication scenarios between two parties can be implemented by first
encoding messages into some states of a physical system which acts as the
physical medium of the communication and then decoding the messages by
measuring the state of the system. We show that already in the simplest
possible scenarios it is possible to detect a definite, unbounded advantage of
quantum systems over classical systems. We do this by constructing a family of
operationally meaningful communication tasks each of which on one hand can be
implemented by using just a single qubit but which on the other hand require
unboundedly larger classical system for classical implementation. Furthemore,
we show that even though with the additional resource of shared randomness the
proposed communication tasks can be implemented by both quantum and classical
systems of the same size, the number of coordinated actions needed for the
classical implementation also grows unboundedly. In particular, no finite
storage can be used to store all the coordinated actions needed to implement
all the possible quantum communication tasks with classical systems. As a
consequence, shared randomness cannot be viewed as a free resource.
