If it was recently thrown upward, it is either still travelling upwards away from the centre of the Earth, or it is falling downwards towards the centre of the Earth.
So its past is kind of irrelevant. At some point if nothing is supporting it, it will fall towards the centre of the Earth.
You don’t need information about its past. Everything you need to know to predict its future state is contained in its current position and velocity at the moment you measure it.
Is there no measurable quantity in the current state of a magnetized ball that would predict its behavior in a magnetic field then?
I'd be surprised if that were the case: seems like it'd violate some sort of "locality", where the future should depend entirely on the present state.
Is it just that there's unidentified present state that's perhaps more easily characterized by knowing the history? Or is it somehow unidentifiable in principle?
Yeah, the current magnetization of the sphere carries all the history of the magnetic fields it's been in in its value.
The hysteresis in the material only means that if you apply 1T to it to magnetise (supposing the ball starts with zero magnetisation) it and then apply 0T to demagnetise it, the remanent field in the ball would be like 0.4T due to hystetesis, you need to apply something like -0.4T to leave it as it started.
A chunk of iron put into a 1μT magnetic field will behave differently depending on what other magnetic fields it has been in recently.