This appears in at 1:28:28. 

Question: "How does your Principle of Computational Equivalence (PCE) 
explain the separation in complexity between a classical cellular 
automaton (CA) and a quantum CA?"

Wolfram: "OK, so... that's a complicated question... So, one question is:
if you take quantum mechanics as we currently know it, and you say,
okay... let's use quantum mechanics... let's just assume that we can make
measurements infinitely quickly and that the standard formalism of quantum
mechanics sort of is an exact description of the actual world, not some
kind of an idealization of the world (as we know that it is, because we
know that in fact when we make measurements... that we have to, for
example, take one... some degree of freedom and amplify it to an almost an
infinite number of degrees of freedom, and things like that... but let's
assume we don't worry about that... let's take the idealization that it's
just... you know... that, as the formalism suggests that we just... do a
measurement and it happens...,) okay, so then the question is what... how
does... how does what I am talking about relate to the computation that
you can get done in the classical case vs. the quantum case. So..., first
thing I should say is that, ultimately, the claim of my PCE is that in our
universe it is in fact not possible to do things that are more
sophisticated than what a classical CA, for example, can do. So, if it
turned out that quantum CAs (or quantum computers of some kind) can do
more sophisticated things than any of these classical CAs then this PCE of
mine would just be wrong." And he smiles (roguish smile) and waits, to see
if the answer thus far is sinking in. At this time the question is refined:

Question: "By more sophisticated do you mean with a better complexity
then, or do you mean in a univers..."

Wolfram: "Okay, so to be more specific... I mean... so one question is:
can it do more than a universal Turing machine, can it break Church's
thesis? Another is: can it do... you know... an NP-complete computation in
polynomial time? Okay... so, on the second issue: I am not so sure! On the
first issue, I'm really... you know... I'm, I'm... I certainly will claim
very strongly... I mean... I certainly believe that in our actual universe
one can't do computations that are more sophisticated than one can get
done by a standard classical Turing machine. Now, having said that there's
already then a sort of technical question which is: if you allow quantum
mechanics... if you allow sort of standard idealizations of quantum
mechanics, which involve complex amplitudes that have an arbitrary number
of bits in them, and things like that, even within the formalism... can
you do computations that are more sophisticated than you can do in a
standard classical universal Turing machine? I think that people generally
feel that probably you can't. But, to show that... would be interesting.
For example, the following is true: if you say that you do computations
just with polynomials, that the only operations that you can have are:
writing down a polynomial, and solving a polynomial. Or, solving a
polynomial inequality or something. Well... then, it turns out that there
is a result from the 1930s, I guess, that it is at least... it's...
that... if you're only working with polynomials and solving polynomials
and so on... then... it's a decidable problem; then, every question you
ask is decidable... it was Tarski's result [...] so, within... if the kind
of primitives you have in your theory would be only polynomial primitives
then it is the case that even using these arbitrary precision amplitudes
in quantum mechanics you couldn't compute something more than what we can
classically compute... but as soon as you allow transcendental functions
for example... that is no longer the case, because for example even with
trigonometric functions you can easily encode an arbitrary diophantine
equation in your... sort of, continuous system, so... if you allow such
functions then you can immediately do computations that cannot be done by
a classical Turing machine and if there was a way to get quantum mechanics
to make use of those transcendental functions, which certainly seems not
impossible that could be... then it would imply that quantum mechanics in
its usual idealization would be capable of doing computations beyond the
Church's thesis limit. So... Then you can ask questions about the NP
completeness level of things: I'm less certain about that... my own
guess... speculation... is that in fact if you try... eventually, unravel
the idealizations that are made in sort of particle [physics] and quantum
mechanics that you will find that in fact you don't succeed in doing
things that are sort of more sophisticated than you could do with a
classical Turing machine type system... but I don't know that for sure!"

That's the end of the answer regarding the relationship between PCE, and 
the classical vs. quantum models of computation. The answer then goes a bit 
further, explaining how NKS is interesting and important even without any 
reference to quantum models of computation, and this last part of Wolfram's 
answer briefly discusses a problem involving a lower bound on a space of 
Turing machines. 

Note: one can also go to the video here: