Abstract: In this seminar, we give a review of the recent development on the quantum Hopf hierarchy and the quantum KdV hierarchy. The quantum Hopf hierarchy (aka the quantum dispersionless KdV hierarchy) was introduced by Eliashberg, and the quantum KdV hierarchy that we consider was constructed by Buryak and Rossi using double ramification cycles on the moduli space of curves. We will mainly focus on the spectral problems of the quantum integrable hierarchies. In particular, we will explain how quasi-modularity naturally appears in this study and Zagier's conjecture.