ILP and number of variables in constraints
There are no hard rules for which formulation would be better especially for mixed-integer problems, however


The time to solve an lp relaxation usually increases with the number of nonzeros in the constraint matrix, but
Tighter lp relaxations usually reduce the number of lp relaxations that need to be solved.


To take your example, suppose you have a variable $y$ that is 1 only if all variables $x_0, x_2, \ldots, x_{n-1}$ have value 1, and all values are binary.  You could write a single constraint
$$ny \le \sum_{i=0}^n x_i$$

or $n$ constraints
$$y \le x_i, \forall i \in \{0, \ldots n-1\}$$

Both sets of constraints will have the same integer-feasible set, but the $n$ constraints have a much smaller continuous-feasible set, so it is probably be much faster to solve.