  We use results from communication complexity, both new and old ones, to prove
lower bounds for unambiguous finite automata (UFAs). We show three results.
  $\textit{Complement:}$ There is a language $L$ recognised by an $n$-state UFA
such that the complement language $\overline{L}$ requires NFAs with
$n^{\tilde{\Omega}(\log n)}$ states. This improves on a lower bound by Raskin.
  $\textit{Union:}$ There are languages $L_1$, $L_2$ recognised by $n$-state
UFAs such that the union $L_1\cup L_2$ requires UFAs with
$n^{\tilde{\Omega}(\log n)}$ states.
  $\textit{Separation:}$ There is a language $L$ such that both $L$ and
$\overline{L}$ are recognised by $n$-state NFAs but such that $L$ requires UFAs
with $n^{\Omega(\log n)}$ states. This refutes a conjecture by Colcombet.
