I think there is a basic misunderstanding here. The Bitcoin blockchain does not store public keys; it stores public key hashes. When you do a transaction, your public key is checked against the public key hash in the blockchain to verify that it's your bitcoin, then the transaction can take place.

Quantum computers aren't particularly good at breaking hashes (which is why hash-based digital signatures are used in quantum resistant cryptography), and will not be able to get a public key from a public key hash anytime soon. The threat is that given a public key, they can calculate the private key and hijack the transaction. The window for doing this is at the time of the transaction and before the transaction is incorporated into the blockchain, because it is only then that the actual public key is exposed. If one were to keep some bitcoin in the address from which a transaction is sent, an attacker, having obtained the public key, could break it at his leisure and eventually access the rest of the bitcoin in that address. But this is already considered very bad practice and nobody does this.

The idea that the entire blockchain is vulnerable to a quantum computer is incorrect, at least for the forseeable future. Grover's algorithm can be used on a quantum computer to break hashes, but it's much less efficient than Shor's algorithm. Therefore ECC will be broken long before quantum computers can threaten hashes.

Or so I understand it.

The Quantum Resistant Ledger (QRL) is a coin that is designed to resist quantum computers. Others are Curecoin and Iota.