Let Mk,m be the space of Laurent polynomials in one variable x+ t1x + . . . tk+mx , where k,m ≥ 1 are fixed integers and tk+m 6= 0. According to B. Dubrovin [11], Mk,m can be equipped with a semi-simple Frobenius structure. In this paper we prove that the corresponding descendent and ancestor potentials of Mk,m (defined as in [16]) satisfy Hirota quadratic equations (HQE for short). Let Ck,m be the orbifold obtained from P by cutting small discs D1 ∼= {|z| ≤ ǫ} and D2 ∼= {|z−1| ≤ ǫ} around z = 0 and z = ∞ and gluing back the orbifolds D1/Zk and D2/Zm in the obvious way. We show that the orbifold quantum cohomology of Ck,m coincides with Mk,m as Frobenius manifolds. Modulo some yet-to-be-clarified details, this implies that the descendent (respectively the ancestor) potential of Mk,m is a generating function for the descendent (respectively ancestor) orbifold Gromov–Witten invariants of Ck,m. There is a certain similarity between our HQE and the Lax operators of the Extended bi-graded Toda hierarchy, introduced by G. Carlet in [7]. Therefore, it is plausible that our HQE characterize the tau-functions of this hierarchy and we expect that the Extended bi-graded Toda hierarchy governs the Gromov–Witten theory of Ck,m.