  The arrival time problem for the free particle in one dimension may be
formulated as the problem of determining a joint probability for the particle
being found on opposite sides of the $x$-axis at two different times. We
explore this problem using a two-time quasi-probability linear in the
projection operators, a natural counterpart of the corresponding classical
problem. We show that it can be measured either indirectly, by measuring its
moments in different experiments, or directly, in a single experiment using a
pair of sequential measurements in which the first measurement is weak (or more
generally, ambiguous). We argue that when positive, it corresponds to a
measurement-independent arrival time probability. For small time intervals it
coincides approximately with the time-averaged current, in agreement with
semiclassical expectations. The quasi-probability can be negative and we
exhibit a number of situations in which this is the case. We interpret these
situations as the presence of `quantumness', in which the arrival time
probability is not properly defined in a measurement-independent manner.
Backflow states, in which the current flows in the direction opposite to the
momentum, are shown to provide an interesting class of examples such
situations. We also show that the quasi-probability is closely linked to a set
of two-time Leggett-Garg inequalities, which test for macroscopic realism.
