  We provide more sample-efficient versions of some basic routines in quantum
data analysis, along with simpler proofs. Particularly, we give a quantum
"Threshold Search" algorithm that requires only $O((\log^2 m)/\epsilon^2)$
samples of a $d$-dimensional state $\rho$. That is, given observables $0 \le
A_1, A_2, ..., A_m \le 1$ such that $\mathrm{tr}(\rho A_i) \ge 1/2$ for at
least one $i$, the algorithm finds $j$ with $\mathrm{tr}(\rho A_j) \ge
1/2-\epsilon$. As a consequence, we obtain a Shadow Tomography algorithm
requiring only $\tilde{O}((\log^2 m)(\log d)/\epsilon^4)$ samples, which
simultaneously achieves the best known dependence on each parameter $m$, $d$,
$\epsilon$. This yields the same sample complexity for quantum Hypothesis
Selection among $m$ states; we also give an alternative Hypothesis Selection
method using $\tilde{O}((\log^3 m)/\epsilon^2)$ samples.
