  We prove that the correlations present in a multipartite quantum state have
an \emph{operational} quantum character as soon as the state does not simply
encode a multipartite classical probability distribution, i.e. does not
describe the joint state of many classical registers. Even unentangled states
may exhibit such \emph{quantumness}, that is pointed out by the new task of
\emph{local broadcasting}, i.e. of locally sharing pre-established
correlations: this task is feasible if and only if correlations are classical
and derive a no-local-broadcasting theorem for quantum correlations. Thus,
local broadcasting is able to point out the quantumness of correlations, as
standard broadcasting points out the quantum character of single system states.
Further, we argue that our theorem implies the standard no-broadcasting theorem
for single systems, and that our operative approach leads in a natural way to
the definition of measures for quantumness of correlations.
