Standard Leggett-Garg inequalities (SLGIs) were formulated for testing incompatibility between the classical world view of macrorealism and quantum mechanics. In recent times, various other formulations such as the Wigner form of Leggett-Garg inequalities (WLGIs), entropic Leggett-Garg inequalities (ELGIs), and the no-signaling-in-time (NSIT) condition have also been proposed. It was also recently argued that
no set of SLGIs can provide the necessary and sufficient conditions for macrorealism, but a suitable conjunction of NSIT conditions provides the same. In this paper, we first provide a comparative study of the various formulations of Leggett-Garg inequalities (LGIs) for testing macrorealism pertaining to the two different unsharp measurements. While the violations of WLGIs are more robust than SLGIs and ELGIs for spin positive operator-valued measures (POVMs), here we demonstrate that for the case of biased POVMs the quantum violations of both SLGIs and ELGIs provide the same robustness as WLGIs. Importantly, the violations of all formulations of LGIs can be achieved for any nonzero value of unsharpness parameter. We have also studied the connection between LGIs and NSIT conditions. Further, we investigate the role of the joint measurability of the POVMs in the violation of LGIs and find that there is
no generic connection.
The Leggett-Garg inequalities are a set of inequalities obeyed by classical systems but violated in quantum theory. Their violation has been taken as evidence that quantum theory lacks a `realistic' formulation. However in addition to realism the derivation of the Leggett-Garg inequalities relies on another, more obscure assumption of `non-invasive measurability.' The significance of this assumption and the consequences for the interpretation of violations of the Leggett-Garg inequalities have been hotly debated. In this paper we present a pedagogical introduction to the issues, focussing on the significance of non-invasive measurability. We give a simple derivation of the Leggett-Garg inequalities paying particular attention to where and when the two assumptions are used, and we give an example of a realist but not non-invasively measurable hidden variable theory which violates them. We also discuss recent attempts to experimentally implement non-invasive measurements via so called `undetectable measurements' and we show that even undetectable measurements are not necessarily non-invasive.
Leggett-Garg inequalities are an important milestone in our quest to bridge the classical-quantum divide. An experimental investigation of these inequalities requires the so called \emph{non-invasive measurements}(NIM). It has become popular to invoke weak measurements as the means of realising NIMs to very good approximation, because of their allegedly low disturbance of systems under measurement. In this note, this is shown to be a myth; it is shown, by simple estimates of errors, that for comparable levels of statistical errors, even the strong or projective measurements can be used. In fact, it is shown that resource-wise, strong measurements are even preferable.