It is known that converting an n-state nondeterministic nested word automaton (a.k.a. input-driven automaton; a.k.a. visibly pushdown automaton) to a corresponding deterministic automaton requires in the worst case 2 ^Θ(n2) states (R. Alur, P. Madhusudan: Adding nesting structure to words, DLT'06). We show that the same worst case 2^Θ(n2) size blow-up occurs when converting a nondeterministic nested word automaton to an unambiguous one, and an unambiguous nested word automaton to a deterministic one. In addition, the methods developed in this paper are used to demonstrate that the state complexity of complementation for nondeterministic nested word automata is 2^Θ(n2), and that the state complexity of homomorphism for deterministic nested word automata is 2^Θ(n2).