  Using the concept of mixable shuffles, we formulate explicitly the quantum
quasi-shuffle product. We also provide a desirable description of the
subalgebra generated by the set of primitive elements of the quantum
quasi-shuffle bialgebra. A braided coalgebra structure which is dual to the
quantum quasi-shuffle in some sense is constructed as well. We use quantum
quasi-shuffle algebras to provide examples of Rota-Baxter algebras and
tridendriform algebras.
