Quantum Composite Hypothesis Testing with Small Error

We present a method of quantum composite hypothesis testing with small error, which enables us to establish quantum lower bounds in nonparametric statistics and high-dimensional functional estimation. This is achieved by introducing trigonometric polynomials into the quantum polynomial method, thereby generalizing the quantum phase-estimation lower bound of Mande and de Wolf (ESA 2023). As applications, we settle the quantum complexities of several problems in property testing and functional estimation with small error: - For ${\ell }_2$-closeness testing, we show that the approach of Luo et al. (*IEEE Trans. Inf. Theory* 2024) is optimal. - For Tsallis entropy estimation, where $q=2$ corresponds to the Gini impurity, we show that the approaches of Buhrman et al. (*Phys. Rev. Lett.* 2001) and Ekert et al. (*Phys. Rev. Lett.* 2002) are optimal for integer $q\ge 2$, and that the approach of Chen et al. (ICALP 2026) is near-optimal for real $q\ge 1.5$. - For pure-state Uhlmann fidelity and trace distance estimation, we show that the approach of Wang (*IEEE Trans. Inf. Theory* 2024) is optimal.