Today was pretty interesting, I realized I had been looking at the problem entirely wrong. My whole plan was the following, use the Lovasz Theta (LT) function to obtain a "good" bound on the maximum eigenvalue of the CSS hamiltonian, and then show that there is a gap between n levels of the NPA hierarchy, and the value of the LT function, meaning that the LT value is tighter than degree n SoS. But I forgot to take into account, that LT is the value of a semi-definite program itself, which is certainly captured by much less than n levels of the SoS hierarchy.. Anyways I am not too bummed about this, because as soon as I realized that, I realized another stupidly obvious fact, I already have a pretty narrow window of what the maximum eigenvalue can be for the CSS hamiltonian! In particular, if the value of the original instance is 1/2 plus delta, then the max eigenvalue of both the Hx summand and the Hz summand will be 1/2 plus delta each. So there is an immediate bound from the triangle inequality on operator norms: the max eigenvalue of H = Hx + Hz is 1 + 2delta. On top of that, consider a 1/2 plus delta satisfiable assignment to the 3XOR instance, which we know is the maximum, this corresponds to a state in the computational basis which will achieve the maximum on the Hz summand, and due to the fact that Z and X are mutually unbiased bases, states in the computational basis (Z) have expectation 0 on the pauli X operator, so the Hx summand will evaluate to 1/2, this is due to the fact that the local summands in Hx for each clause are Cx = (I + XiXjXk)/2, so if Tr(XiXjXk p) = 0, then Tr(Cx p) = 1/2. So in total, we know that the max eigenvalue is going to be in the range [1 + delta, 1 + 2*delta]. That's pretty good, I am curious about what states might beat 1 + delta, are they going to be NLTS? Anyways, thats the mini result of the day, after that I relentlessly tried to analyze the lovasz theta function of the underlying conflict graph but I am having a REALLY hard time. It's one of those objects where there is no framework for actually thinking about it. It has a structure similar to a johnson scheme graph but those are constructed out of complete graphs on n choose k vertices. The original graph that this conflict graph is constructed from is not complete, and it has this annoying bipartite property due to the fact that all terms in the same pauli basis commute, so I can't seem to find any previous work on bounding the LT function for these graphs. It seems I will have to generalize something or find a different way to estimate this value, oh yea and the reason I still care about the LT function is I think its a good first step to estimating what NPA_o(n)(H) is going to be. Maybe it will help build intuition for what the pseudodistribution will look like.