  For a commutative algebra the shuffle product is a morphism of complexes. We
generalize this result to the quantum shuffle product, associated to a class of
non-commutative algebras (for example all the Hopf algebras). As a first
application we show that the Hochschild-Serre identity is the dual statement of
our result. In particular, we extend this identity to Hopf algebras. Secondly,
we clarify the construction of a class of quasi-Hopf algebras.
