Bcrypt is not asymmetric.
Couldn't you make it asymmetric if you introduced another alogrithm into the mix in a smart manner?
Are you suggesting something like Lamport's signature, which builds a digital signature algorithm from the SHA256? It's not clear where the Bcrypt's advatange is here, Bcrypt is computationally intensive, but that doesn't necessarily imply QC resistance.
Well you could have a schedule and raise the computational power it takes to get the hash as time goes by, this will not only solve the quantum computer problem but all other advances in technology as well.
SHA256 is the least problem you would have to worry about, without QC, see here https://i.imgur.com/fYFBsqp.jpg
And if QC is invented, it cannot just circumvent the SHA256, instead it will weaken it to the SHA128 level using Grover's algorithm, closer but still nearly insurmountable.
If you want to replace the hash function to counter the threat of QC, take care that if QC would not be widely deployed but concentrated in a few places, the slowdown introduced by bcrypt may affect the performance of other non-QC miners even more, and force them to quit mining, the scrypt however, which throttles the hashrate with the memory bottleneck, may prove to be useful.
From what I understand the whole concern is with scrypt rather than SHA256, SHA256 would take way, way too long to break.
Depending on the definition of "break", in some ways it's not breakable at all.
