  We consider two-variable first-order logic $\text{FO}^2$ and its quantifier
alternation hierarchies over both finite and infinite words. Our main results
are forbidden patterns for deterministic automata (finite words) and for
Carton-Michel automata (infinite words). In order to give concise patterns, we
allow the use of subwords on paths in finite graphs. This concept is formalized
as subword-patterns. For certain types of subword-patterns there exists a
non-deterministic logspace algorithm to decide their presence or absence in a
given automaton. In particular, this leads to $\mathbf{NL}$ algorithms for
deciding the levels of the $\text{FO}^2$ quantifier alternation hierarchies.
This applies to both full and half levels, each over finite and infinite words.
Moreover, we show that these problems are $\mathbf{NL}$-hard and, hence,
$\mathbf{NL}$-complete.
