On boundedness of characteristic class via quasi-morphism

In the present paper, we characterize the second bounded characteristic classes of foliated bundles in terms of the homogeneous quasi-morphisms on the universal covering of the structure group. As its application, we prove the boundedness and unboundedness of obstruction classes of (contact) Hamiltonian fibrations and the non-triviality of the second bounded cohomology of the group of Hamiltonian diffeomorphisms of any closed symplectic manifolds.