  By weakly measuring the polarization of a photon between two strong
polarization measurements, we experimentally investigate the correlation
between the appearance of anomalous values in quantum weak measurements, and
the violation of realism and non-intrusiveness of measurements. A quantitative
formulation of the latter concept is expressed in terms of a Leggett-Garg
inequality for the outcomes of subsequent measurements of an individual quantum
system. We experimentally violate the Leggett-Garg inequality for several
measurement strengths. Furthermore, we experimentally demonstrate that there is
a one-to-one correlation between achieving strange weak values and violating
the Leggett-Garg inequality.
