  We construct an example of a non-trivial homogeneous quasimorphism on the
group of Hamiltonian diffeomorphisms of the two and four dimensional quadric
hypersurfaces which is continuous with respect to both the $C^0$-metric and the
Hofer metric. This answers a variant of a question of Entov--Polterovich--Py
which is one of the open problems listed in the monograph of McDuff--Salamon.
Throughout the proof, we make extensive use of the idea of working with
different coefficient fields in quantum cohomology rings. As a by-product of
the arguments in the paper, we answer a question of Polterovich--Wu regarding
quasimorphisms on the group of Hamiltonian diffeomorphisms of the complex
projective plane and prove some intersection results about Lagrangians in the
four dimensional quadric hypersurface.
