Questions about the polynomial-time hierarchy are studied. In particular, the questions, “Does the polynomial-time hierarchy collapse?” and “Is the union of the hierarchy equal to PSPACE?” are considered, along with others comparing the union of the hierarchy with certain probabilistic classes. In each case it is shown that the answer is “yes” if and only if for every sparse set S, the answer is “yes” when the classes are relativized to S if and only if there exists a sparse set S such that the answer is “yes” when the classes are relativized to S. Thus, in each case the question is answered if it is answered for any arbitrary sparse oracle set.

Long and Selman first proved that the polynomial-time hierarchy collapses if and only if for every sparse set S, the hierarchy relative to S collapses. This result is re-proved here by a different technique.