  Recent work of Acharya et al. (NeurIPS 2019) showed how to estimate the
entropy of a distribution $\mathcal D$ over an alphabet of size $k$ up to
$\pm\epsilon$ additive error by streaming over $(k/\epsilon^3) \cdot
\text{polylog}(1/\epsilon)$ i.i.d. samples and using only $O(1)$ words of
memory. In this work, we give a new constant memory scheme that reduces the
sample complexity to $(k/\epsilon^2)\cdot \text{polylog}(1/\epsilon)$. We
conjecture that this is optimal up to $\text{polylog}(1/\epsilon)$ factors.
