  Quark-hadron duality implies that a process described in terms of quark loops
should be the hadronic amplitude when averaged over a sufficient number of
states. Ambiguities associated with the notion of quark hadron duality can be
made arbitrarily small for highly excited mesons at large $N_c$. QCD is
expected to form a string like description at large $N_c$ yielding an
exponentially increasing Hagedorn spectrum for high mass. It is shown that in
order to reconcile quantum-hadron duality with a Hagedorn spectrum, the
magnitude of individual coupling constants between high-lying mesons in a
typical decay process must be characteristically larger than the average of the
coupling constants to mesons with nearby masses. The ratio of the square of the
average coupling to the average of the coupling squared (where the average is
over mesons with nearby masses) drops {\it exponentially} with the mass of the
meson. Scenarios are discussed by which such a high precision cancellation can
occur.
