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General relativity makes clear predictions about the spacetime geometryaround black holes. In the near future, new facilities will have the capabilityto explore the metric around SgrA$^*$, the supermassive black hole candidate atthe Center of our Galaxy, and open a new window to test the Kerr black holehypothesis. In this paper, we compute light curves and images associated withcompact emission regions (hot spots) orbiting around Kerr and non-Kerr blackholes. We study how the analysis of the properties of the radiation emitted bya hot spot can be used to test the Kerr nature of SgrA$^*$. We find that thesole observation of the hot spot light curve can at most constrain acombination of the black hole spin and of possible deviations from the Kerrsolution. This happens because the same orbital frequency around a Kerr blackhole can be found for a non-Kerr object with a different spin parameter. Secondorder corrections in the light curve due to the background geometry aretypically too small to be identified. While the observation of the hot spotcentroid track can potentially bound possible deviations from the Kerrsolution, that is out of reach for the near future VLTI instrument GRAVITY. TheKerr black hole hypothesis could really be tested in the case of the discoveryof a radio pulsar in a compact orbit around SgrA$^*$. Radio observations ofsuch a pulsar would provide precise estimates of the mass and the spin ofSgrA$^*$, and the combination of these measurements (probing the weak field)with the hot spot light curve information (probing the strong field) mayconstrain/find possible deviations from the Kerr solution with quite goodprecision.

Testing the nature of the supermassive black hole candidate in SgrA* with light curves and images of hot spots