  This paper studies the complexity classes QZK and HVQZK of problems having a
quantum computational zero-knowledge proof system and an honest-verifier
quantum computational zero-knowledge proof system, respectively. The results
proved in this paper include: (a) HVQZK = QZK, (b) any problem in QZK has a
public-coin quantum computational zero-knowledge proof system, (c) any problem
in QZK has a quantum computational zero-knowledge proof system of perfect
completeness, and (d) any problem in QZK has a three-message public-coin
quantum computational zero-knowledge proof system of perfect completeness with
arbitrarily small constant error in soundness. All the results above are
unconditional and do not rely any computational assumptions. For the classes
QPZK, HVQPZK, and QSZK of problems having a quantum perfect zero-knowledge
proof system, an honest-verifier quantum perfect zero-knowledge proof system,
and a quantum statistical zero-knowledge proof system, respectively, the
following new properties are proved: (e) HVQPZK = QPZK, (f) any problem in QPZK
has a public-coin quantum perfect zero-knowledge proof system, (g) any problem
in QSZK has a quantum statistical zero-knowledge proof system of perfect
completeness, and (h) any problem in QSZK has a three-message public-coin
quantum statistical zero-knowledge proof system of perfect completeness with
arbitrarily small constant error in soundness. It is stressed that our proofs
are direct and do not use complete promise problems or those equivalents. This
gives a unified framework that works well for all of quantum perfect,
statistical, and computational zero-knowledge proofs, and enables us to prove
properties even on the computational and perfect zero-knowledge proofs for
which no complete promise problems are known.
