  We examine the power of statistical zero knowledge proofs (captured by the
complexity class SZK) and their variants. First, we give the strongest known
relativized evidence that SZK contains hard problems, by exhibiting an oracle
relative to which SZK (indeed, even NISZK) is not contained in the class UPP,
containing those problems solvable by randomized algorithms with unbounded
error. This answers an open question of Watrous from 2002 [Aar]. Second, we
"lift" this oracle separation to the setting of communication complexity,
thereby answering a question of G\"o\"os et al. (ICALP 2016). Third, we give
relativized evidence that perfect zero knowledge proofs (captured by the class
PZK) are weaker than general zero knowledge proofs. Specifically, we exhibit
oracles relative to which SZK is not contained in PZK, NISZK is not contained
in NIPZK, and PZK is not equal to coPZK. The first of these results answers a
question raised in 1991 by Aiello and H{\aa}stad (Information and Computation),
and the second answers a question of Lovett and Zhang (2016). We also describe
additional applications of these results outside of structural complexity.
  The technical core of our results is a stronger hardness amplification
theorem for approximate degree, which roughly says that composing the
gapped-majority function with any function of high approximate degree yields a
function with high threshold degree.
