  Recently, a new class of so-called \emph{hierarchical thresholding
algorithms} was introduced to optimally exploit the sparsity structure in joint
user activity and channel detection problems. In this paper, we take a closer
look at the user detection performance of such algorithms under noise and
relate its performance to the classical block correlation detector with
orthogonal signatures. More specifically, we derive a lower bound for the
diversity order which, under suitable choice of the signatures, equals that of
the block correlation detector. Surprisingly, in specific parameter settings
non-orthogonal pilots, i.e. pilots where (cyclically) shifted versions
interfere with each other, outperform the block correlation detector.
Altogether, we show that, in wide parameter regimes, the hierarchical
thresholding detectors behave like the classical correlator with improved
detection performance but operate with much less required pilot subcarriers. We
provide mathematically rigorous and easy to handle formulas for numerical
evaluations and system design. Finally, we evaluate our findings with numerical
examples and show that, in a practical parameter setting, a classical pilot
channel can accommodate up to three advanced pilot channels with the same
performance.
