  We introduce the "moment rank" and "unitary rank" of numerical sequences,
close relatives of linear-recursive order. We show that both parameters can be
characterized by a broad set of criteria involving moments of measures, types
of recurrence relations, Hankel matrix factorizations, Waring rank, analytic
properties of generating functions, and algebraic properties of polynomial
ideals. In the process, we solve the "complex finite-atomic" and "integral
finite-atomic" moment problems: which sequences arise as the moments of a
finite-atomic complex-/integer-valued measures on $\mathbb{C}$?
