  It is known that three fundamental questions regarding local Hamiltonians --
approximating the ground state energy (the Local Hamiltonian problem),
simulating local measurements on the ground space (APX-SIM), and deciding if
the low energy space has an energy barrier (GSCON) -- are $\mathsf{QMA}$-hard,
$\mathsf{P}^{\mathsf{QMA}[log]}$-hard and $\mathsf{QCMA}$-hard, respectively,
meaning they are likely intractable even on a quantum computer. Yet while
hardness for the Local Hamiltonian problem is known to hold even for
translationally-invariant systems, it is not yet known whether APX-SIM and
GSCON remain hard in such "simple" systems. In this work, we show that the
translationally invariant versions of both APX-SIM and GSCON remain
intractable, namely are $\mathsf{P}^{\mathsf{QMA}_{\mathsf{EXP}}}$- and
$\mathsf{QCMA}_{\mathsf{EXP}}$-complete, respectively. Each of these results is
attained by giving a respective generic "lifting theorem" for producing
hardness results. For APX-SIM, for example, we give a framework for "lifting"
any abstract local circuit-to-Hamiltonian mapping $H$ (satisfying mild
assumptions) to hardness of APX-SIM on the family of Hamiltonians produced by
$H$, while preserving the structural and geometric properties of $H$ (e.g.
translation invariance, geometry, locality, etc). Each result also leverages
counterintuitive properties of our constructions: for APX-SIM, we "compress"
the answers to polynomially many parallel queries to a QMA oracle into a single
qubit. For GSCON, we give a hardness construction robust against highly
non-local unitaries, i.e. even if the adversary acts on all but one qudit in
the system in each step.
