  Macrorealism (MR) is the world view that certain quantities may take definite
values at all times irrespective of past or future measurements and may be
experimentally falsified via the Leggett-Garg (LG) inequalities. We put this
world view to the test for systems described by a continuous variable $x$ by
seeking LG violations for measurements of a dichotomic variable $Q =
\textrm{sign}(x)$, in the case of gaussian initial states in a quantum harmonic
oscillator. Extending our earlier analysis [C. Mawby and J. J. Halliwell, Phys.
Rev. A 105, 022221 (2022)] we find analytic expressions for the temporal
correlators. An exploration of parameter space reveals significant regimes in
which the two-time LG inequalities are violated, and likewise at three and four
times. To obtain a physical picture of the LG violations, we exploit the
continuous nature of the underlying position variable and analyse the relevant
quantum-mechanical currents, Bohm trajectories, and Wigner function. Further,
we extend the analysis LG tests using coherent state projectors, thermal
coherent states, and squeezed states.
