Can a full-scale quantum computer own Bitcoin blockchain? Yes.
Nothing to discuss below this point, IMO.
Well, if full-scale quantum computers are used in mining, they would simply "up" the difficulty, in the same way that the introduction of ASICS did. You would have miners with quantum computers "competing" for blocks and all the rest left behind, like the CPU miners were left behind by ASIC miners.
Of course, the problem with PoW as a cryptographic security mechanism is that it is a quite ridiculous "security" in the sense that the work to be done by an attacker is comparable to the work done by the "good guys" ; and in the case of technological evolution, the work to be done by an attacker is MUCH LESS than the total work done by all the good guys (work done as in economic cost of work done).
For instance, with ASIC miners, it is very easy to overdo all the work that miners did the first few years of bitcoin, for much less cost than these miners did spend on it.
So if ever there is a quantum computer that can mine, miners should use it directly, and the "old part of the chain" will be easy to redo with the quantum computer at hand ; but competition between quantum computers will still make the "head" of the chain ; like ASICS now make the head of the chain, but the old tail is relatively easy to do over.