Leggett–Garg inequality
The Leggett–Garg inequality, named for Anthony James Leggett and Anupam Garg, is a mathematical inequality fulfilled by all macrorealistic physical theories. Here, macrorealism (macroscopic realism) is a classical worldview defined by the conjunction of two postulates: In quantum mechanics. In quantum mechanics, the Leggett–Garg inequality is violated, meaning that the time evolution of a system cannot be understood classically. The situation is similar to the violation of Bell's inequalities in Bell test experiments which plays an important role in understanding the nature of the Einstein–Podolsky–Rosen paradox. Here quantum entanglement plays the  central role. Two-state example. The simplest form of the Leggett–Garg inequality derives from examining a system that has only two possible states. These states have corresponding measurement values formula_1. The key here is that we have measurements at two different times, and one or more times between the first and last measurement. The simplest example is where the system is measured at three successive times formula_2. Now suppose, for instance, that there is a perfect correlation formula_3 of 1 between times formula_4 and formula_5. That is to say, that for N realisations of the experiment, the temporal correlation  reads We look at this case in some detail. What can be said about what happens at time formula_7? Well, it is possible that formula_8, so that if the value of formula_9 at formula_4 is formula_11, then it is also formula_11 for both times formula_13 and formula_5. It is also quite possible that formula_15, so that the value of formula_9 at formula_17 is flipped twice, and so has the same value at formula_18 as it did at formula_4. So, we can have both formula_20 and formula_21 anti-correlated as long as we have formula_21 and formula_23 anti-correlated. Yet another  possibility is that there is no correlation between formula_20 and formula_21. That is we could have formula_26. So, although it is known that if formula_1 at formula_28 it must also be formula_11 at formula_5, the value at formula_13 may as well be determined by the toss of a coin. We define formula_32 as formula_33. In these three cases, we have formula_34 and formula_35, respectively. All that was for 100% correlation between times formula_36 and formula_37. In fact, for any correlation between these times formula_38. To see this, we note that It is easily seen that for every realisation formula_40, the  term in the parentheses must be less than or equal to unity, so that the result for the average is also less than (or equal to) unity. If we have four distinct times rather than three, we have formula_41 and so on. These are the Leggett–Garg inequalities. They express the relation between the temporal correlations of formula_42 and the correlations between successive times in going from the start to the end. In the derivations above, it has been assumed that the quantity Q, representing the state of the system, always has a definite value (macrorealism per se) and that  its measurement at a certain time does not change this value nor its subsequent evolution (noninvasive measurability). A violation of the Leggett–Garg inequality implies that at least one of these two assumptions fails. Experimental violations. One of the first proposed experiments for demonstrating a violation of macroscopic realism employs superconducting quantum interference devices. There, using Josephson junctions, one should be able to prepare macroscopic superpositions of left and right rotating macroscopically large electronic currents in a superconducting ring. Under sufficient suppression of decoherence one should be able to demonstrate a violation of the Leggett–Garg inequality. However, some criticism  has been raised concerning the nature of indistinguishable electrons in a Fermi sea. A criticism of some other proposed experiments on the Leggett–Garg inequality is that they do not really show a violation of macrorealism because they are essentially about measuring spins of individual particles. In 2015 Robens "et al." demonstrated an experimental violation of the Leggett–Garg inequality using superpositions of positions instead of spin with a massive particle. At that time, and so far up until today, the Cesium atoms employed in their experiment represent the largest quantum objects which have been used to experimentally test the  Leggett–Garg inequality. The experiments of Robens "et al." as well as Knee "et al.", using ideal negative measurements, also avoid a second criticism (referred to as “clumsiness loophole”) that has been directed to previous experiments using measurement protocols that could be interpreted as invasive, thereby conflicting with postulate 2. Several other experimental violations have been reported, including in 2016 with neutrino particles using the MINOS dataset. Brukner and Kofler have also demonstrated that quantum violations can be found for arbitrarily large "macroscopic" systems. As an alternative to quantum decoherence, Brukner and Kofler are proposing a solution of the quantum -to-classical transition in terms of "coarse-grained" quantum measurements under which usually no violation of the Leggett–Garg inequality can be seen anymore. Experiments proposed by Mermin and Braunstein and Mann would be better for testing macroscopic realism, but warns that the experiments may be complex enough to admit unforeseen loopholes in the analysis. A detailed discussion of the subject can be found in the review by Emary et al. Related inequalities. The four-term Leggett–Garg inequality can be seen to be similar to the CHSH inequality. Moreover, "equalities" were proposed by Jaeger "et al." Leggett–Garg inequality