Given a projective variety X, one can ask if there exist curves on X that have smaller numerical invariants than those coming from slicing by some ample divisors. For a general complete intersection X of large degrees, we show that there are no curves on X of smaller degree, nor are there curves of asymptotically smaller gonality. This verifies a folklore conjecture on the degrees of subvarieties of complete intersections as well as a conjecture of Bastianelli--De Poi--Ein--Lazarsfeld--Ullery. Joint work with Ben Church and Junyan Zhao.