  Quantum metrology bears a great promise in enhancing measurement precision,
but is unlikely to become practical in the near future. Its concepts can
nevertheless inspire classical or hybrid methods of immediate value. Here, we
demonstrate NOON-like photonic states of m quanta of angular momentum up to
m=100, in a setup that acts as a "photonic gear", converting, for each photon,
a mechanical rotation of an angle {\theta} into an amplified rotation of the
optical polarization by m{\theta}, corresponding to a "super-resolving" Malus'
law. We show that this effect leads to single-photon angular measurements with
the same precision of polarization-only quantum strategies with m photons, but
robust to photon losses. Moreover, we combine the gear effect with the quantum
enhancement due to entanglement, thus exploiting the advantages of both
approaches. The high "gear ratio" m boosts the current state-of-the-art of
optical non-contact angular measurements by almost two orders of magnitude.
