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In this paper, we show how quantum modular forms naturally arise in theergodic theory of circle rotations. Working with the classical Birkhoff sum$S_N(\alpha)=\sum_{n=1}^N (\{ n \alpha \}-1/2)$, we prove that the maximum andthe minimum as well as certain exponential moments of $S_N(r)$ as functions of$r \in \mathbb{Q}$ satisfy a direct analogue of Zagier's continuity conjecture,originally stated for a quantum invariant of the figure-eight knot. As acorollary, we find the limit distribution of $\max_{0 \le N<M} S_N(\alpha)$ and$\min_{0 \le N<M} S_N(\alpha)$ with a random $\alpha \in [0,1]$.

Limit laws of maximal Birkhoff sums for circle rotations via quantum modular forms