It seems that many results in complexity hold assuming PH doesn't collapse to the second or third levels.  At these low levels, I have some intuition about the collapse not occurring since additional quantifiers seem to add power not already inherent in the lower levels, but higher levels escape my understanding.  Basically, are there any known results that separate levels of PH assuming other complexity theoretic separations?  Is there any intuition that PH doesn't collapse to some finite level other than its generalization from P vs. NP and logical hierarchies?  
Collapsing PH Implications
cc.complexity theory