A QC factors the current public key algorithms in constant time. More qubits just means you can factor bigger keys. How big that constant is will depend on how the QC works.
Sending privately to a miner would help, but you'd end up with a very centralized system since you would want to send to the biggest miner, to minimize the time until that miner produces a block. You can't have miners sharing the transaction, even privately among themselves, since if they did that then one of them could have a QC and you wouldn't have any way to know who stole your money.
