  We study the determinisation and unambiguisation problems of weighted
automata over the rational field: Given a weighted automaton, can we determine
whether there exists an equivalent deterministic, respectively unambiguous,
weighted automaton? Recent results by Bell and Smertnig show that the problem
is decidable, however they do not provide any complexity bounds. We show that
both problems are in PSPACE for polynomially-ambiguous weighted automata.
