There may be some difference in accounting. The recursion depth of Blelloch's recursive function you linked to is O(log n), but it uses an operation that cannot be implemented in constant time: partitioning the elements into those greater and less than the pivot. I'd guess that operation is itself O(log n), which would account for the O(log^2 n).
By "in some sense", do you mean it can be parallelized to O(1) time as long as you don't have to actually build up the data structures that result from partitioning (e.g. arrays)? Then it's not clear to me why this assumption is applicable to quicksorting an array. Doesn't it call for some non-obvious representation of intermediate results?
But isn't recursion depth the same as depth if each operation in your recursive function is constant time (except for the recursing bits)? Thus you can look at recursive depth as computing depth modulo the bits you compute in your function at a fixed level. I was speculating that partitioning would require O(log n) parallel steps. Can you give details or provide a link to achieving it constant depth?
