  We explore potential quantum speedups for the fundamental problem of testing
the properties of closeness and $k$-wise uniformity of probability
distributions.
  Closeness testing is the problem of distinguishing whether two
$n$-dimensional distributions are identical or at least $\varepsilon$-far in
$\ell^1$- or $\ell^2$-distance. We show that the quantum query complexities for
$\ell^1$- and $\ell^2$-closeness testing are $O(\sqrt{n}/\varepsilon)$ and
$O(1/\varepsilon)$, respectively, both of which achieve optimal dependence on
$\varepsilon$, improving the prior best results of Gily\'en and Li (2020).
  $k$-wise uniformity testing is the problem of distinguishing whether a
distribution over $\{0, 1\}^n$ is uniform when restricted to any $k$
coordinates or $\varepsilon$-far from any such distributions. We propose the
first quantum algorithm for this problem with query complexity
$O(\sqrt{n^k}/\varepsilon)$, achieving a quadratic speedup over the
state-of-the-art classical algorithm with sample complexity
$O(n^k/\varepsilon^2)$ by O'Donnell and Zhao (2018). Moreover, when $k = 2$ our
quantum algorithm outperforms any classical one because of the classical lower
bound $\Omega(n/\varepsilon^2)$.
  All our quantum algorithms are fairly simple and time-efficient, using only
basic quantum subroutines such as amplitude estimation.
