  What advantage do \emph{sequential} procedures provide over batch algorithms
for testing properties of unknown distributions? Focusing on the problem of
testing whether two distributions $\mathcal{D}_1$ and $\mathcal{D}_2$ on
$\{1,\dots, n\}$ are equal or $\epsilon$-far, we give several answers to this
question. We show that for a small alphabet size $n$, there is a sequential
algorithm that outperforms any batch algorithm by a factor of at least $4$ in
terms sample complexity. For a general alphabet size $n$, we give a sequential
algorithm that uses no more samples than its batch counterpart, and possibly
fewer if the actual distance $TV(\mathcal{D}_1, \mathcal{D}_2)$ between
$\mathcal{D}_1$ and $\mathcal{D}_2$ is larger than $\epsilon$. As a corollary,
letting $\epsilon$ go to $0$, we obtain a sequential algorithm for testing
closeness when no a priori bound on $TV(\mathcal{D}_1, \mathcal{D}_2)$ is given
that has a sample complexity
$\tilde{\mathcal{O}}(\frac{n^{2/3}}{TV(\mathcal{D}_1, \mathcal{D}_2)^{4/3}})$:
this improves over the $\tilde{\mathcal{O}}(\frac{n/\log n}{TV(\mathcal{D}_1,
\mathcal{D}_2)^{2} })$ tester of \cite{daskalakis2017optimal} and is optimal up
to multiplicative constants. We also establish limitations of sequential
algorithms for the problem of testing identity and closeness: they can improve
the worst case number of samples by at most a constant factor.
