Recent Discoveries in Quantum Hamiltonian Complexity
 Kitaev proved that kLH is in QMA for k>=1 and QMAhard for k>=5.
 Oliveira and Terhal showed that LH with Hamiltonians restricted to nearest neighbor interaction on a 2D grid is still QMAcomplete.
 Aharonov, Gottesman, Irani and Kempe showed that 2LH with nearest neighbor interactions on the line is also QMA complete if the local system have dimension of at least 12.
 Cubitt and Montanaro established a quantum variant of Schaefers Dichotomy Theorem.
 Bravyi defined Quantum kSAT in which all local constraints are positive semidefinite, and the question is whether the ground state energy is zero.
 Aharonov and Eldar proved that approximating the ground state energy of commuting local Hamiltonians on good locallyexpanding graphs is in NP.
 Gharibian, Landau, Shin and Wang showed that the commuting variant of the Stoquastic kSAT problem is in NP for logarithmic k and any constant d.
 Yan and Bacon showed that LH with all commuting terms are products of Pauli operators is in P.
 Schuch and Verstraete showed that Hubbard Model has QMA hardness.
 Bravyi has given a Fully Polynomial Randomized Approximation Scheme for approximating the partition function of the transverse field Ising model.
 Baharonas works showed that finding a ground state and computing the magnetic partition function of an Ising spin glass in a nonuniform magnetic field are NP hard tasks.
 Whites discovered his celebrated Density Matrix Renormalization Group.
 Verstraete and Cirac Projected entangled pair states
 Schuch, Wolf, Verstraete and Cirac found that MPS and MERA networks can be efficiently contracted.
 Arad, Kitaev, Landau, Vazirani showed that MPS with sublinear bond dimension suffices to approximate the ground state.
 A combinatorial proof improving on Hastings result for the frustrationfree case was later given by Aharonov, Arad, Landau and Vazarani.
 Cubitt, Perez, Garcia, Wolf showed that determining whether a translationalinvariant, nearestneighbor Hamiltonian on a 2D square lattice is gapped is undecidable.
 Gosset, Terhal, and Vershynina showed how to perform universal adiabatic quantum computation using spacetime circuittoHamiltonian construction.
 Fitzsimons and Vidick gave a multiprover interactive proof system for the Local Hamiltonian problem involving a constant number of entangled provers.
 Chubb and Flamia extended the works of Landau, Vazirani and Vidick and Huang to give a polynomial time algorithm for approximating ground space projectors of gapped 1D Hamiltonians with degenerate groun spaces.
 Gharibian and Sikora showed that given a local Hamiltonian and two ground states, is there a sequence of local unitaries mapping between the two states.
 Ge and Eisert showed that in two and higher dimensions, it is not general true that an area law for the Renyi entanglement entropy implies the ability to faithfully describe a quantum manybody state by an efficient tensor network.
 Aharonov showed that a generalized version of the area law fails to hold.
 Movassah and Shor showed that a generalization of Bravyis spin1 model to integer spins chains yields a power law violation of the area law.
 Brandao and Cramer showed that the exponential decay in the specific heat capacity at low temperatures yields an area law for lowenergy states
 Marien proved that the stability of the area law for entanglement entropy in quantum spin systems in the setting of adiabatic and quasiadiabatic evolutions.
