Baseball’s roster rules are almost unbelievably complicated and MLB does not actually publish then, but basically:
There is no minimum amount of time a player has to be brought up for, but there is a minimum amount of time he must spend in the minors once sent down.
Since to be brought to the major leagues a player must be on the 40-man roster, which always includes the 25 men in the majors plus 15 men in the minors or on the d15-day DL, there is a practical limit to how much you can shuffle guys up and down without having a bunch of them trapped in the minors
Once a player is on his team’s 40-man roster, his team has three “option years” during which he can be moved up and down at will. After that he must clear waivers before being sent to the minors, or be designated for assignment, which means he either clears waivers, is traded, or is released.
After 5 years of major league service a player cannot be sent to the minor leagues without his consent.
So, there’s a bunch of interrelated rules that limit how much you can use minor league transactions to keep your bullpen fresh. In fact teams DO do this; they’ll bring a few young guys up for a short period of time to have fresh arms handy, or bring a guy up to make one start. But you can’t do it too much, and only with less experienced players.