  Dubrovin has shown that the spectrum of the quantization (with respect to the
first Poisson structure) of the dispersionless Korteweg-de Vries (KdV)
hierarchy is given by shifted symmetric functions; the latter are related by
the Bloch-Okounkov Theorem to quasimodular forms on the full modular group. We
extend the relation to quasimodular forms to the full quantum KdV hierarchy
(and to the more general quantum Intermediate Long Wave hierarchy). These
quantum integrable hierarchies have been defined by Buryak and Rossi in terms
of the Double Ramification cycle in the moduli space of curves. The main tool
and conceptual contribution of the paper is a general effective criterion for
quasimodularity.
