  Generating ground states of any local Hamiltonians seems to be impossible in
quantum polynomial time. In this paper, we give evidence for the impossibility
by applying an argument used in the quantum-computational-supremacy approach.
More precisely, we show that if ground states of any $3$-local Hamiltonians can
be approximately generated in quantum polynomial time with postselection, then
${\sf PP}={\sf PSPACE}$. Our result is superior to the existing findings in the
sense that we reduce the impossibility to an unlikely relation between
classical complexity classes. We also discuss what makes efficiently generating
the ground states hard for postselected quantum computation.
