We give constraints on when the n-fold cyclic branched cover Σn(L) of a strongly quasipositive link L can be an L-space. In particular we show that if K is a non-trivial L-space knot and Σn(K) is an L-space then (1) n≤5; (2) if n = 4 or 5 then K is the torus knot T(2,3); (3) if n = 3 then K is either T(2,3) or T(2,5), or K is hyperbolic and has the same Alexander polynomial as T(2,5); (4) if n = 2 then ΔK(t) is a non-trivial product of cyclotomic polynomials. (This is joint work with Michel Boileau and Steve Boyer.)