Given a projective variety X, it is always covered by curves obtained by taking the intersection with a linear subspace. We study whether there exist curves on X that have smaller numerical invariants than those of the linear slices. If X is a general complete intersection 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 on measures of irrationality for complete intersections. This is joint work with Nathan Chen and Junyan Zhao.