  Reconstruction of population histories is a central problem in population
genetics. Existing coalescent-based methods, like the seminal work of Li and
Durbin (Nature, 2011), attempt to solve this problem using sequence data but
have no rigorous guarantees. Determining the amount of data needed to correctly
reconstruct population histories is a major challenge. Using a variety of tools
from information theory, the theory of extremal polynomials, and approximation
theory, we prove new sharp information-theoretic lower bounds on the problem of
reconstructing population structure -- the history of multiple subpopulations
that merge, split and change sizes over time. Our lower bounds are exponential
in the number of subpopulations, even when reconstructing recent histories. We
demonstrate the sharpness of our lower bounds by providing algorithms for
distinguishing and learning population histories with matching dependence on
the number of subpopulations. Along the way and of independent interest, we
essentially determine the optimal number of samples needed to learn an
exponential mixture distribution information-theoretically, proving the upper
bound by analyzing natural (and efficient) algorithms for this problem.
