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The Leggett-Garg (LG) inequalities were introduced, as a temporal parallel of the Bell inequalities, to test macroscopic realism -- the view that a macroscopic system evolving in time possesses definite properties which can be determined without disturbing the future or past state. The talk will begin with a review of the LG framework. Unlike the Bell inequalities, the original LG inequalities are only a necessary condition for macrorealism, and are therefore not a decisive test. I argue, for the case of measurements of a single dichotomic variable Q, that when the original four three-time LG inequalities are augmented with a set of twelve two-time inequalities also of the LG form, Fine's theorem applies and these augmented conditions are then both necessary and sufficient [1]. A comparison is carried out with the alternative necessary and sufficient conditions for macrorealism  based on no-signaling in time conditions which ensure that all probabilities for Q at one and two times are independent of whether earlier or intermediate measurements are made. I argue that the two tests differ in their implementation of the key requirement of non-invasive measurability so are testing different notions of macrorealism, and these notions are elucidated.  I also describe some alternative protocols which achieve non-invasiveness, one involving continuous measurement of the velocity conjugate to Q [2], which was recently implemented in an experiment at IQC, the other involving a modification of the standard ideal negative measurement protocol [3].

[1] J.J.Halliwell, Phys Rev A96, 012121 (2017); A93, 022123 (2016); arxiv:1811.10408.

[2]  J.J.Halliwell, Phys. Rev. A94, 052114 (2016).

[3] J.J.Halliwell, Phys. Rev. A99, 022119 (2019).