#### PhD Preliminary

The Quantum PCP Conjecture (QPCP) claims that estimating ground-state energies of local Hamiltonians to constant precision is hard for quantum computers. If true, QPCP implies the existence of local Hamiltonians with non-trivial low-energy space properties, and a recent trend has been to construct (or conjecture) such Hamiltonians independently of QPCP. This proposal will focus on topics surrounding QPCP and one such implication, the No Low-energy Sampleable States (NLSS) Conjecture. The results accomplished thus-far include two necessary consequences of both QPCP and the NLSS Conjecture: constructions of No Low-energy Stabilizer States (NLCS) Hamiltonians and No Low-energy Almost-Clifford States (NLACS) Hamiltonians. After detailing why QPCP implies certain Hamiltonians we will examine the explicit construction of NLCS/NLACS Hamiltonians. Will also see how the construction can be adapted to yield simultaneous NLACS/No Low-energy Trivial States (NLTS) Hamiltonians, another prerequisite of QPCP. We will end with proposed research directions in both Hamiltonian complexity and quantum error correction which have connections to QPCP.