We consider random simple temporal graphs in which every edge of the complete graph \(K_n\) appears once within the time interval [0, 1] independently and uniformly at random. Our main result is a sharp threshold on the size of any maximum \(\delta \) -clique (namely a clique with edges appearing at most \(\delta \) apart within [0, 1]) in random instances of this model, for any constant  \(\delta \) . In particular, using the probabilistic method, we prove that the size of a maximum \(\delta \) -clique is approximately \(\frac{2\log {n}}{\log {\frac{1}{\delta }}}\) with high probability (whp). We note that, even though the random simple temporal graph contains \(\varTheta (n^2)\) overlapping \(\delta \) -windows, which (when viewed separately) correspond to different random instances of the Erdős-Rényi random graphs model, the size of the maximum \(\delta \) -clique in the former model and the maximum clique size of the latter are approximately the same. Furthermore, we show that the minimum interval containing a \(\delta \) -clique is \(\delta -o(\delta )\) whp. We use this result to show that any polynomial time algorithm for \(\delta \) -Temporal Clique is unlikely to have very large probability of success.