  A well-motivated conjecture states that the open topological string partition
function on toric geometries in the Nekrasov-Shatashvili limit is annihilated
by a difference operator called the quantum mirror curve. Recently, the complex
structure variables parameterizing the curve, which play the role of
eigenvalues for related operators, were conjectured to satisfy a quantization
condition non-perturbative in the NS parameter $\hbar$. Here, we argue that
this quantization condition arises from requiring single-valuedness of the
partition function, combined with the requirement of smoothness in the
parameter $\hbar$. To determine the monodromy of the partition function, we
study the underlying difference equation in the framework of exact WKB.
