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Quantum periods appear in many contexts, from quantum mechanics to local
mirror symmetry. They can be described in terms of topological string free
energies and Wilson loops, in the so-called Nekrasov-Shatashvili limit. We
consider the trans-series extension of the holomorphic anomaly equations
satisfied by these quantities, and we obtain exact multi-instanton solutions
for these trans-series. Building on this result, we propose a unified
perspective on the resurgent structure of quantum periods. We show for example
that the Delabaere-Pham formula, which was originally obtained in quantum
mechanical examples, is a generic feature of quantum periods. We illustrate our
general results with explicit calculations for the double-well in quantum
mechanics, and for the quantum mirror curve of local $\mathbb{P}^2$.
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