  We consider a problem introduced by Feige, Gamarnik, Neeman, R\'acz and
Tetali [2020], that of finding a large clique in a random graph $G\sim
G(n,\frac{1}{2})$, where the graph $G$ is accessible by queries to entries of
its adjacency matrix. The query model allows some limited adaptivity, with a
constant number of rounds of queries, and $n^\delta$ queries in each round.
With high probability, the maximum clique in $G$ is of size roughly $2\log n$,
and the goal is to find cliques of size $\alpha\log n$, for $\alpha$ as large
as possible. We prove that no two-rounds algorithm is likely to find a clique
larger than $\frac{4}{3}\delta\log n$, which is a tight upper bound when
$1\leq\delta\leq \frac{6}{5}$. For other ranges of parameters, namely,
two-rounds with $\frac{6}{5}<\delta<2$, and three-rounds with $1\leq\delta<2$,
we improve over the previously known upper bounds on $\alpha$, but our upper
bounds are not tight. If early rounds are restricted to have fewer queries than
the last round, then for some such restrictions we do prove tight upper bounds.
