Perhaps not "obvious", but QM (pretty much by definition) only alters time complexity of processes, not space complexity. Any process, it doesn't matter what we are talking about. And the consequences of qualitative changes in time complexity for any abstract process is well understood. I don't need to know what QM is or even how it works. We don't even have to be talking about QM per se.
If QM modified space complexity, then one could definitely make an interesting argument. But even in the most abstract theory, QM is not capable of having this property for a myriad of reasons. Therfore, its impact is reducible to a very simple and predictable case of having some (unknown) effect on time complexity.
As to computational theory, I don't know, although I don't think a turing machine will "get us there" (due to Godel's Incompleteness Theorem, a la Penrose).
Penrose makes his argument for a case with very restrictive assumptions, and is not even applicable to most of the models actually used in standard theory. In this sense, Penrose has built a theoretical strawman for himself. And in fact, mathematicians have proven that the arguments he uses are invalid for the classes of model normally used by the core theory folks. Penrose's arguments ONLY apply to axiomatic models, but much practical theory uses purely non-axiomatic models. People who invoke Penrose have to understand that it is limited in applicability to axiomatic models, while most current models are non-axiomatic and have been for some time. In this sense, Penrose is a non-argument -- theoretical apples and oranges.
Suffice it to say that Penrose's argument is not applicable to the universal model I am assuming, as his premises are orthogonal to mine. And I would point out that most other mathematicians who work in core theory are assuming the same basic model parameters as I am. Penrose may have a point of some type, but it has no relation to the work that most other people are doing.
Penrose has convinced me that current theory is at least incomplete and that there is need for a "new physics", if you will.
There is nothing in physics or any of the rest of the sciences that begins to explain the whole human being. Consciousness is an ever-present, universal characteristic, not an epiphenomenon.
Have you read, or has anyone here read, The Conscious Universe by Dean Radin. It's science and rigorous, not bunk, but it will make Materialist scientists uncomfortable.