  We investigate SU(3) gauge theories in four dimensions with Nf fundamental
fermions, on a lattice using the Wilson fermion. Clarifying the vacuum
structure in terms of Polyakov loops in spatial directions and properties of
temporal propagators using a new method "local analysis", we conjecture that
the "conformal region" exists together with the confining region and the
deconfining region in the phase structure parametrized by beta and K, both in
the cases of the large Nf QCD within the conformal window (referred as
Conformal QCD) with an IR cutoff and small Nf QCD at T/Tc>1 with Tc being the
chiral transition temperature (referred as High Temperature QCD).
  Our numerical simulation on a lattice of the size 16^3 x 64 shows the
following evidence of the conjecture. In the conformal region we find the
vacuum is the nontrivial Z(3) twisted vacuum modified by non-perturbative
effects and temporal propagators of meson behave at large t as a power-law
corrected Yukawa-type decaying form. The transition from the conformal region
to the deconfining region or the confining region is a sharp transition between
different vacua and therefore it suggests a first order transition both in
Conformal QCD and in High Temperature QCD.
  Within our fixed lattice simulation, we find that there is a precise
correspondence between Conformal QCD and High Temperature QCD in the temporal
propagators under the change of the parameters Nf and T/Tc respectively. In
particular, we find the correspondence between Conformal QCD with Nf = 7 and
High Temperature QCD with Nf=2 at T ~ 2 Tc being in close relation to a meson
unparticle model. From this we estimate the anomalous mass dimension gamma* =
1.2 (1) for Nf=7. We also show that the asymptotic state in the limit T/Tc -->
infty is a free quark state in the Z(3) twisted vacuum.
