  The polylogarithmic time hierarchy structures sub-linear time complexity. In
recent work it was shown that all classes $\tilde{\Sigma}_{m}^{\mathit{plog}}$
or $\tilde{\Pi}_{m}^{\mathit{plog}}$ ($m \in \mathbb{N}$) in this hierarchy can
be captured by semantically restricted fragments of second-order logic. In this
paper the descriptive complexity theory of polylogarithmic time is taken
further showing that there are strict hierarchies inside each of the classes of
the hierarchy. A straightforward consequence of this result is that there are
no complete problems for these complexity classes, not even under polynomial
time reductions. As another consequence we show that the polylogarithmic time
hierarchy itself is strict.
