  We obtain worst case performance guarantees for $p=2$ and $3$ QAOA for MAXCUT
on uniform 3-regular graphs. Previous work by Farhi et al obtained a lower
bound on the approximation ratio of $0.692$ for $p=1$. We find a lower bound of
$0.7559$ for $p=2$, where worst case graphs are those with no cycles $\leq 5$.
This bound holds for any 3 regular graph evaluated at particular fixed
parameters. We conjecture a hierarchy for all $p$, where worst case graphs have
with no cycles $\leq 2p+1$. Under this conjecture, the approximation ratio is
at least $0.7924$ for all 3 regular graphs and $p=3$. In addition, using a
simple indistinguishability argument we find an upper bound on the worst case
approximation ratio for all $p$, which indicates classes of graphs for which
there can be no quantum advantage for at least $p<6$.
