  The black hole in the center of the Milky Way, Sgr A*, has the largest
mass-to-distance ratio among all known black holes in the Universe. This
property makes Sgr A* the optimal target for testing the gravitational no-hair
theorem. In the near future, major developments in instrumentation will provide
the tools for high-precision studies of its spacetime via observations of
relativistic effects in stellar orbits, in the timing of pulsars, and in
horizon-scale images of its accretion flow. We explore here the prospect of
measuring the properties of the black-hole spacetime using all these three
types of observations. We show that the correlated uncertainties in the
measurements of the black-hole spin and quadrupole moment using the orbits of
stars and pulsars are nearly orthogonal to those obtained from measuring the
shape and size of the shadow the black hole casts on the surrounding emission.
Combining these three types of observations will, therefore, allow us to assess
and quantify systematic biases and uncertainties in each measurement and lead
to a highly accurate, quantitative test of the gravitational no-hair theorem.
