Homogeneous quasimorphisms, $C^0$-topology and Lagrangian intersection

We construct an example of a non-trivial homogeneous quasimorphism on the group of Hamiltonian diffeomorphisms of the $2$- and $4$-dimensional quadric which is continuous with respect to both $C^0$-topology 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. One of the key ideas is to work with quantum cohomology rings with different coefficient fields which might be of independent interest. As another application of this idea, we answer a question of Polterovich-Wu. Some by-products about Lagrangian intersection and the existence of Lagrangian submanifolds that are diffeomorphic but not Hamiltonian isotopic for the $4$-dimensional quadric are also discussed.