Sharon must have broken up with her boyfriend yesterday.
Sharon must have been broken up with her boyfriend yesterday.
Sharon must have broken up with her boyfriend yesterday.
Sharon must have been broken up with her boyfriend yesterday.
The same question was asked and answered the other day, but since that seems to have been deleted...
Must have, in this context, means that the speaker believes something to be true based on the available evidence.
To break up with a romantic partner means to end the relationship with them.
Sharon must have broken up with her boyfriend yesterday.
This means that the speaker believes Sharon ended her relationship with her boyfriend yesterday. Out of context it's hard to say if the important point is the breakup, or that it happened yesterday.
Sharon must have been broken up with her boyfriend yesterday.
This doesn't mean that the breakup happened yesterday, it means that yesterday Sharon and her boyfriend were in a state of already being broken up, after having broken up previously (whether the breakup occurred the day before, the week before, or whatever).
Perhaps yesterday the speaker saw Sharon out with some other guy, and knowing that Sharon wouldn't cheat on her boyfriend concluded that she and her boyfriend must have been [in a state of having] broken up.
Or, some couples have an on again/off again relationship: they keep breaking up, then getting back together. If that is the case with Sharon, then been broken up...yesterday means that yesterday was one of the times when Sharon and her boyfriend were not a couple because their most recent breakup was before that.
They're very similar.
The first sentence is basically assuming this (if you reword it):
Yesterday, Sharon broke up with her boyfriend.
It's assuming that the action of breaking up happened yesterday.
In this context, it seems that the second sentence would be said if you thought Sharon had broken up yesterday, but weren't exactly sure. So something like this:
Sharon must have been broken up with her boyfriend by then, because she was acting so strangely yesterday.
Here's a good link for a question on the grammar in your second sentence: https://english.stackexchange.com/questions/151284/how-can-i-use-must-have-been