  The main theme of this paper is to use toric degeneration to produce distinct
homogeneous quasimorphisms on the group of Hamiltonian diffeomorphisms. We
focus on the (complex $n$-dimensional) quadric hypersurface and the del Pezzo
surfaces, and study two classes of distinguished Lagrangian submanifolds that
appear naturally in a toric degeneration, namely the Lagrangian torus which is
the monotone fiber of a Lagrangian torus fibration, and the Lagrangian spheres
that appear as vanishing cycles. For the quadrics, we prove that the group of
Hamiltonian diffeomorphisms admits two distinct homogeneous quasimorphisms and
derive some superheaviness results. Along the way, we show that the toric
degeneration is compatible with the Biran decomposition. This implies that for
$n=2$, the Lagrangian fiber torus (Gelfand--Zeitlin torus) is Hamiltonian
isotopic to the Chekanov torus, which answers a question of Y. Kim. We give
applications to $C^0$-symplectic topology which include the
Entov--Polterovich--Py question for the quadric hypersurface. We also prove
analogous results for the del Pezzo surfaces.
