  We show that Cutting Planes (CP) proofs are hard to find: Given an
unsatisfiable formula $F$,
  1) It is NP-hard to find a CP refutation of $F$ in time polynomial in the
length of the shortest such refutation; and
  2)unless Gap-Hitting-Set admits a nontrivial algorithm, one cannot find a
tree-like CP refutation of $F$ in time polynomial in the length of the shortest
such refutation.
  The first result extends the recent breakthrough of Atserias and M\"uller
(FOCS 2019) that established an analogous result for Resolution. Our proofs
rely on two new lifting theorems: (1) Dag-like lifting for gadgets with many
output bits. (2) Tree-like lifting that simulates an $r$-round protocol with
gadgets of query complexity $O(\log r)$ independent of input length.
