Open Problems
 Bose–Hubbard CGW14 proof on different geometries. The proof of [CGW14] is for symmetric $0$–$1$ graphs with at most one selfloop. What about simple graphs? Lattices? No self–loops?
 What are the complexities of the HaF instances of Hamiltonians?
 What is the complexity of varying Ising Model instances on different geometries?
 What are the complexity classifications of Hamiltonians with interaction strengths of the form $J = O(1)$?
 The general definition of the models admits algebraic locality and not geometrical locality. It is therefore natural to ask what the complexity is for interaction strengths of the form, $J_{ij} = i  j^{\alpha}$.
 The BPP algorithm presented by Bravyi [Bra15] for calculating the partition function of the ferromagnetic TIM is not practical. Is there an improved version that can offer practicality?
 The advent of guided local Hamiltonian instances begs the question as to whether similar results can be found for the likes of the Fermi–Hubbard, Bose–Hubbard, and Electronic Structure Hamiltonian.
 Are hybrid interaction systems of interest and can their complexity also be defined?
 What is the complete impact of guiding states on the complexity of local Hamiltonians?

Can a robust framework of local Hamiltonians with restricted ground states be defined?
Guiding states have been proposed that have a semi–classical strucutre. Recently, local Hamiltonians with succinct ground states have been proposed. Furthermore, finding extremal product states for local Hamiltonians has been studied. Is there a better framework with which we can discuss these ideas and find possible links?

What alternate ideas exist for circuit–to–Hamiltonian constructions?
Are there good constructions without the need for a clock? Furthermore, are there constructions that are more physically motivated?
 Is it #Phard to approximate partition functions at real temperature?