  Let $L$ be a oriented link such that $\Sigma_n(L)$, the $n$-fold cyclic cover
of $S^3$ branched over $L$, is an L-space for some $n \geq 2$. We show that if
either $L$ is a strongly quasipositive link other than one with Alexander
polynomial a multiple of $(t-1)^{2g(L) + (|L|-1)}$, or $L$ is a quasipositive
link other than one with Alexander polynomial divisible by $(t-1)^{2g_4(L) +
(|L|-1)}$, then there is an integer $n(L)$, determined by the Alexander
polynomial of $L$ in the first case and the Alexander polynomial of $L$ and the
smooth $4$-genus of $L$, $g_4(L)$, in the second, such that $n \leq n(L)$. If
$K$ is a strongly quasipositive knot with monic Alexander polynomial such as an
L-space knot, we show that $\Sigma_n(K)$ is not an L-space for $n \geq 6$, and
that the Alexander polynomial of $K$ is a non-trivial product of cyclotomic
polynomials if $\Sigma_n(K)$ is an L-space for some $n = 2, 3, 4, 5$. Our
results allow us to calculate the smooth and topological 4-ball genera of, for
instance, quasi-alternating quasipositive links. They also allow us to classify
strongly quasipositive alternating links and $3$-strand pretzel links.
