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 = ω(\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 Θ(\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 = Ω(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.