** Abstract: **

In both query and communication complexity, we give separations between the class NISZK,
containing those problems with non-interactive statistical zero knowledge proof systems,
and the class UPP, containing those problems with randomized algorithms with unbounded error.
These results significantly improve on earlier query separations of Vereschagin [Ver95] and
Aaronson [Aar12] and earlier communication complexity separations of Klauck [Kla11] and
Razborov and Sherstov [RS10]. In addition, our results imply an oracle relative to which
the class NISZK is not contained in PP. This answers an open question of Watrous from
2002 [Aar]. The technical core of our result 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.

Using our techniques, we also give oracles relative to which the following two separations
hold: honest-verifier perfect zero knowledge (HVPZK) is not contained in its complement
(coHVPZK), and SZK (indeed, even NISZK) is not contained in PZK (indeed, even HVPZK).
Along the way, we show that HVPZK is contained in PP in a relativizing manner.

We prove a number of implications of these results, which may be of independent
interest outside of structural complexity. Specifically, our oracle separation
implies that certain parameters of the Polarization Lemma of Sahai and Vadhan [SV03]
cannot be much improved in a black-box manner. Additionally, it implies new lower
bounds for property testing algorithms with error probability arbitrarily close to 1/2.
Finally, our results imply that two-message protocols in the streaming interactive
proofs model of Cormode et al. [CTY11] are surprisingly powerful in the sense that,
with just logarithmic cost, they can compute functions outside of UPP^CC.

** Versions: **

- Mansucript [ECCC]