Inflation v.s. Big Bang It is said that that transition or certain quantum rotor field, and “rolling” down into the valley (of some quantum potential), is what causes inflation to come to an end, and create the hot Big Bang.
Thus this statement from https://medium.com/starts-with-a-bang/the-multiverse-is-inevitable-and-were-living-in-it-311fd1825c6 suggests that

inflation first, then big bang.
see also discussions: Is the Big Bang defined as before or after Inflation?

My question is that how do we contrast the process of inflation and big bang, in terms of a full time evolution?

*

*for example, from which time scale do we have inflation, then from which time scale do we have big bang? and for which time scale and for how long do we have something "????" in between?


inflation --> "????" --> big bang



*Is inflation or is big bang more quantum mechanical?  Is inflation or is big bang more classical or semi-classical?

 A: The options as I understand it are


*

*Hot Big Bang with no inflation

*Hot Big Bang and then a period of inflation

*warmish modest bang followed by a period of inflation which results in conditions of a hot Big Bang
All of 1,2,3 are thoroughly quantum mechanical through and through in their earliest stages, when fast processes on small distance scales (Planck time, Planck distance) have to be accounted for. All can then be treated by classical general relativity once the evolution has passed from such an early stage to a more gradual evolution.
If there was inflation then it would have been at energy scales $10^{15}$ GeV or more, at times of order $10^{-35}$ s. Early models suggested it took place at this sort of time, i.e. after the early Planck era, but models have subsequently investigated a wider range of possibilities extending back to earlier times.
The evidence for inflation is indirect; it consists largely in the spectrum of the anisotropies in the CMB. That same spectrum could have come about without inflation, as I understand it, if the earliest part of a Big Bang (i.e. the Planck era) had resulted in the right kind of early fluctuations in density and curvature. However I am not an expert on that so perhaps others could comment. The other evidence for inflation is that we don't find a high density of magnetic monopoles which are predicted by models of particle physics at these very high energy scales, but here we are very much in unknown territory. The evidence against inflation is that it requires one to assume very special initial conditions, such that overall it is not clear whether it has explained more things than it has assumed.
To conclude, then, inflationary cosmology remains a research program rather than an established result; the presentations of this on Wikipedia etc. tend to imply it is much more settled than in fact it is.
The way inflation can result in conditions of a hot Big Bang is, to put it simply, that there is a transfer of large amounts of energy from the inflaton field to all the other fields at the end of an inflationary episode, and if this transfer happens just the right way, and so as to impart momentum just the right way, then one has subsequently the evolution of an expanding universe described by the Einstein field equation, from a hot dense state, starting from the time $t_{(\mbox{end of inflation})}$ which is the time one might ordinarily call $10^{-33}$ s or something like that. 
