  We present two new results about exact learning by quantum computers. First,
we show how to exactly learn a $k$-Fourier-sparse $n$-bit Boolean function from
$O(k^{1.5}(\log k)^2)$ uniform quantum examples for that function. This
improves over the bound of $\widetilde{\Theta}(kn)$ uniformly random
\emph{classical} examples (Haviv and Regev, CCC'15). Additionally, we provide a
possible direction to improve our $\widetilde{O}(k^{1.5})$ upper bound by
proving an improvement of Chang's lemma for $k$-Fourier-sparse Boolean
functions. Second, we show that if a concept class $\mathcal{C}$ can be exactly
learned using $Q$ quantum membership queries, then it can also be learned using
$O\left(\frac{Q^2}{\log Q}\log|\mathcal{C}|\right)$ \emph{classical} membership
queries. This improves the previous-best simulation result (Servedio and
Gortler, SICOMP'04) by a $\log Q$-factor.
