I am concerned that quantum computing could potentially brute-force all existing wallets. However, I am unsure whether internet connection speed would impose a limitation on such attacks.
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Yeah, like connection speed will be a limitation! Why would you even need an internet connection for brute-forcing private keys? Its not like quantum computers are downloading them off the web.
The real issue with quantum computing and crypto isnt about internet speed; its that quantum computers
could potentially break the math that secures it. But thats not related just to crypto currencies - almost all cryptography encryptions will be in jeopardy, including your bank account.