**Authors:** Richard Beigel, Richard Chang, and Mitsunori Ogiwara

**Abstract:**
Chang and Kadin have shown that if the difference hierarchy over NP
collapses to level *k*, then the polynomial hierarchy (PH) is
equal the *k*th level of the difference hierarchy over
Σ_{2}^{p}. We simplify their proof and obtain a
slightly stronger conclusion: If the difference hierarchy over NP
collapses to level *k*, then PH collapses to _{(k - 1)-tt}^{NP})^{NP}*k* - 1^{NP} and an unlimited
number of queries to a set in NP. We also extend the result to
classes other than NP: For any class *C* that has
≤_{m}^{p}-complete sets and is closed under
≤^{p}_{conj}- and
≤_{m}^{NP}-reductions (alternatively, closed under
≤^{p}_{disj}- and
≤_{m}^{co-NP}reductions), if the difference
hierarchy over *C* collapses to level *k*, then
PH^{C} = _{(k
- 1)-tt}^{NP})^{C}_{=}P is closed under
≤^{p}_{disj}- and
≤_{m}^{co-NP}-reductions. Consequently, if the
difference hierarchy over C_{=}P collapses to level *k*
then PH^{PP} ^{C=P})_{(k -
1)-tt}^{NP})^{PP}

Finally we consider two ways of relativizing the bounded query class
P_{k-tt}^{NP}: the restricted relativization
P_{k-tt}^{NPC}, and the full
relativization (P_{k-tt}^{NP})^{C}. If *C* is
NP-hard, then we show that the two relativizations are different
unless PH^{C} collapses.