On semisimplicity of quantum cohomology of P^1-orbifolds

For a P^1-orbifold C, we prove that its big quantum cohomology is generically
semisimple. As a corollary, we verify a conjecture of Dubrovin for orbi-curves. We also
show that the small quantum cohomology of C is generically semisimple iff C is Fano, i.e.
it has positive orbifold Euler characteristic.