Hey!
I was thinking - is there a way to describe all the minterms of given function, without describing the whole truth-table?

if the input is long (lets say > 5) the truth table is getting too big, what makes this method pretty hard (i’m not even talking about the fact that SOP/POS are not compact representation of the function).

You would need to specify how the function is represented.
If the function is represented by its truth table, then this representation is already “long”, and you have a “budget” for the minterms (simpy because you have to read the truth table itself).
If a function is represented by a Boolean formula, then it is “hard” to decide if the formula is satisfiable (i.e., if the function is not constant zero).
By “hard” I mean, this is an NP-Complete problem. So we do not know of an efficient way (i.e. polynomial time) to decide.