Anomalous weak values and the violation of a multiple-measurement Leggett-Garg inequality

Quantum mechanics presents peculiar properties that, on the one hand, have been the subject of several theoretical and experimental studies about its very foundations and, on the other hand, provide tools for developing new technologies, the so-called quantum technologies. The nonclassicality pointed out by Leggett-Garg inequalities has represented, with Bell inequalities, one of the most investigated subject. In this letter we study the connection of Leggett-Garg inequalities with a new emerging field of quantum measurement, the weak values. In particular, we perform an experimental study of the four-time correlators Legget-Garg test, by exploiting single and sequential weak measurements performed on heralded single photons. We show violation of a four-parameters Leggett-Garg inequality in different experimental conditions, demonstrating an interesting connection between Leggett-Garg inequality violation and anomalous weak values.