We investigate the expressive power of quantifier alternation hierarchy of first-order logic over words. This hierarchy includes the classes Σi (sentences having at most i blocks of quantifiers starting with an ∃) and BΣi (Boolean combinations of Σi 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 BΣ2 and Σ3, by relying on a decision problem called separation, and solving it for Σ2. The contribution of this paper is a generalization of these results to the setting of infinite words: we solve separation for Σ2 and Σ3, and obtain decidable characterizations of BΣ2 and Σ3 as consequences.