  In this paper, we characterize the second bounded characteristic classes of
foliated bundles in terms of the non-descendible quasi-morphisms on the
universal covering of the structure group. As its application, we study the
boundedness of obstruction classes for (contact) Hamiltonian fibrations and
show the non-existence of foliated structures on some Hamiltonian fibrations.
Moreover, for any closed symplectic manifold, we show the non-triviality of the
second bounded cohomology group of the Hamiltonian diffeomorphism group.
