  We study the planted clique problem in which a clique of size k is planted in
an Erdos-Renyi graph G(n,1/2) and one is interested in recovering this planted
clique. It is widely believed that it exhibits a statistical-computational gap
when computational efficiency is equated with the existence of polynomial time
algorithms. We study this problem under a more fine-grained computational lens
and consider the following two questions.
  1. Do there exist sublinear time algorithms for recovering the planted
clique?
  2. What is the smallest running time any algorithm can hope to have?
  We show that because of a well known clique-completion property, very
elementary sublinear time recovery algorithms do indeed exist for clique sizes
k = {\omega}(\sqrt{n}). This points to a qualitatively stronger
statistical-computational gap. The planted clique recovery problem can be
solved without even looking at most of the input above the {\Theta}(\sqrt{n})
threshold and cannot be solved by any efficient algorithm below it.
  A running time lower bound for the recovery problem follows easily from the
results of [RS19], and this implies our recovery algorithms are optimal
whenever k = {\Omega}(n^{2/3}). However, for k = o(n^{2/3}) there is a gap
between our algorithmic upper bound and the information-theoretic lower bound
implied by [RS19].
  With some caveats, we show stronger detection lower bounds based on the
Planted Clique Conjecture for a natural but restricted class of algorithms. The
key idea is to relate very fast sublinear time algorithms for detecting large
planted cliques to polynomial time algorithms for detecting small planted
cliques.
