  We study the univariate moment problem of piecewise-constant density
functions on the interval $[0,1]$ and its consequences for an inference problem
in population genetics. We show that, up to closure, any collection of $n$
moments is achieved by a step function with at most $n-1$ breakpoints and that
this bound is tight. We use this to show that any point in the $n$th
coalescence manifold in population genetics can be attained by a piecewise
constant population history with at most $n-2$ changes. Both the moment cones
and the coalescence manifold are projected spectrahedra and we describe the
problem of finding a nearest point on them as a semidefinite program.
