We argue that near-future detections of gravitational waves from merging black hole binaries can test a long-standing proposal, originally due Bekenstein and Mukhanov, that the areas of black hole horizons are quantized in integer multiples of the Planck area times an $mathcal O(1)$ dimensionless constant $alpha$. This condition quantizes the frequency of radiation that can be absorbed or emitted by a black hole. If this quantization applies to the ring down gravitational radiation emitted immediately after a black hole merger, a single measurement consistent with the predictions of classical general relativity would rule out most or all (depending on the spin of the hole) of the extant proposals in the literature for the value of $alpha$. A measurement of two such events for final black holes with substantially different spins would rule out the proposal for any $alpha$. If the modification of general relativity is confined to the near-horizon region within the holes light ring and does not affect the initial ring down signal, a detection of echoes with characteristic properties could still confirm the proposal.
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