  We show that the polygamous nature of multi-party quantum correlations can be
characterized in a {\em stronger} form based on Tsallis $q$-entropy and
$q$-expectation value. By considering the amount of entanglement that can be
distributed in multi-party systems, we establish a class of strong polygamy
inequalities of multi-party entanglement in terms of Tsallis $q$-entropy and
$q$-expectation for $q \geq 1$. Our new class of inequalities is in fact
tighter than the usual polygamy inequalities of multi-party entanglement, and
the tightness is explicitly illustrated by an example. Moreover, our new class
of inequalities is concerned with the $q$-expected entanglement distributed
between a single party and any possible subsets of the rest parties whereas the
usual polygamy inequality only considers the entanglement between a single
party and another. We further establish the equivalence between strong polygamy
inequalities of quantum entanglement and quantum discord distributed in
multi-party quantum systems.
