  We investigate the quantifier alternation hierarchy in first-order logic on
finite words. Levels in this hierarchy are defined by counting the number of
quantifier alternations in formulas. We prove that one can decide membership of
a regular language to the levels $\mathcal{B}\Sigma_2$ (boolean combination of
formulas having only 1 alternation) and $\Sigma_3$ (formulas having only 2
alternations beginning with an existential block). Our proof works by
considering a deeper problem, called separation, which, once solved for lower
levels, allows us to solve membership for higher levels.
