1. 1. If for any k> 1, the class NPkV collapses into the class NP(k − 1)V, then the polynomial hierarchy collapses to Σ₂^P.

2. 2. If the converse of the above result is true, then any proof of this converse cannot relativize. We exhibit an oracle relative to which the polynomial hierarchy collapses to P^NP, but the output-multiplicity hierarchy is strict.

3. 3. Relative to a random oracle, the output-multiplicity hierarchy is strict. This result is in contrast to the still open problem of the strictness of the polynomial hierarchy relative to a random oracle.

In introducing the technique for the third result we prove a related result of interest: relative to a random oracle UP ≠ NP.