  We investigate the expressive power of quantifier alternation hierarchy of
first-order logic over words. This hierarchy includes the classes ${\Sigma}_i$
(sentences having at most $i$ blocks of quantifiers starting with an $\exists$)
and $\mathcal{B}{\Sigma}_i$ (Boolean combinations of ${\Sigma}_i$ sentences).
So far, this expressive power has been effectively characterized for the lower
levels only. Recently, a breakthrough was made over finite words, and decidable
characterizations were obtained for $\mathcal{B}{\Sigma}_2$ and ${\Sigma}_3$,
by relying on a decision problem called separation, and solving it for
${\Sigma}_2$. The contribution of this paper is a generalization of these
results to the setting of infinite words: we solve separation for ${\Sigma}_2$
and ${\Sigma}_3$, and obtain decidable characterizations of
$\mathcal{B}{\Sigma}_2$ and ${\Sigma}_3$ as consequences.
