  We extend topological string methods in order to perform WKB approximations
for quantum mechanical problems with higher order potentials efficiently. This
requires techniques for the evaluation of the relevant quantum periods for
Riemann surfaces beyond genus one. The basis of these quantum periods is fixed
using the leading behaviour of the classical periods. The full expansion of the
quantum periods is obtained using a system of Picard-Fuchs like operators for a
sequence of integrals of meromorphic forms of the second kind. Discrete
automorphisms of simple higher order potentials allow to view the corresponding
higher genus curves as covering of a genus one curve. In this case the quantum
periods can be alternatively obtained using the holomorphic anomaly solved in
the holomorphic limit within the ring of quasi modular forms of a congruent
subgroup of SL$(2,\mathbb{Z})$ as we check for a symmetric sextic potential.
