  When geometric structures on surfaces are determined by the lengths of
curves, it is natural to ask: which curves' lengths do we really need to know?
It is a result of Duchin--Leininger--Rafi that any flat metric induced by a
unit-norm quadratic differential is determined by its marked simple length
spectrum. We generalize the notion of simple curves to that of q-simple curves,
for any positive integer q, and show that the lengths of q-simple curves
suffice to determine a non-positively curved Euclidean cone metric induced by a
q-differential.
