  We answer two questions regarding the sum-of-squares for the SYK model left
open in Ref. 1, both of which are related to graphs. First (a "limitation"), we
show that a fragment of the sum-of-squares, in which one considers commutation
relations of degree-$4$ Majorana operators but does not impose any other
relations on them, does not give the correct order of magnitude bound on the
ground state energy. Second (a "separation"), we show that the graph invariant
$\Psi(G)$ defined in Ref. 1 may be strictly larger than the independence number
$\alpha(G)$. The invariant $\Psi(G)$ is a bound on the norm of a Hamiltonian
whose terms obey commutation relations determined by the graph $G$, and it was
shown that $\alpha(G)\leq \Psi(G) \leq \vartheta(G)$, where $\vartheta(\cdot)$
is the Lovasz theta function. We briefly discuss the case of $q\neq 4$ in the
SYK model. Separately, we define a problem that we call the quantum knapsack
problem.
