  Quantum theory provides rules governing much of the microscopic world, and
among its counter-intuitive consequences are correlations that exceed the
bounds from local, classical theories. In two-level quantum systems - qubits -
unitary dynamics theoretically limit these spatiotemporal quantum correlations,
called Bell/Clauser-Horn-Shimony-Holt or Leggett-Garg inequalities, to
$2\sqrt{2}$ or 1.5 respectively. Experiments with state-of-the-art qubits have
approached the spatial, Bell and temporal, Leggett-Garg quantum correlation
bounds. Here, using a dissipative, trapped $^{40}$Ca$^+$ ion governed by a
two-level, non-Hermitian Hamiltonian, we observe temporal correlation values up
to 1.703(4) for the Leggett-Garg parameter $K_3$, clearly exceeding the
hitherto inviolable L\"{u}der's bound of 1.5. These excesses occur across the
exceptional point of the parity-time symmetric Hamiltonian responsible for the
qubit's non-unitary, coherent dynamics. Distinct evolution speeds for antipodal
qubit states, which violate the unified (Mendelstam-Tamm or Margolus-Levitin)
bound $\tau_{\textrm{QSL}}$ for the transit time based on quantum speed limit,
result in the super-quantum $K_3$ values observed over a wide parameter range.
Our results demonstrate that post-selected, coherent dynamics of non-Hermitian
Hamiltonians pave the way for enhanced quantum correlations that exceed
protocols based on unitary or dissipative dynamics.
