  The Event Horizon Telescope is a global very-long baseline interferometer
capable of probing potential deviations from the Kerr metric, which is believed
to provide the unique description of astrophysical black holes. Here we report
an updated constraint on the quadrupolar deviation of Sagittarius A* within the
context of a radiatively inefficient accretion flow model in a quasi-Kerr
background. We also simulate near-future constraints obtainable by the
forthcoming eight-station array and show that in this model already a one-day
observation can measure the spin magnitude to within $0.005$, the inclination
to within $0.09^\circ$, the position angle to within $0.04^\circ$, and the
quadrupolar deviation to within $0.005$ at $3\sigma$ confidence. Thus, we are
entering an era of high-precision strong gravity measurements.
