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 ^ { \ast } 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 .