Minimal requirements for the proof of work

I have started learning the bitcoin blockchain from the technical perspective and while lots of things do make sense to me immediately, I am somewhat confused about some details about the proof of work concept. Namely, in the current implementation to successfuly mine the block one is required to find the number (nonce) such that the hash(block, nonce) returns n leading zeros. This n is chosen by the network in such a way that hashpower of the network produces one block every T = 10 minutes. My question is: what if this system had lower T?

My guess is: the lower the T the higher is the risk - but the risk of what and why? For example, which undesirable situation is much likely to happen in case T is just 1 second? The answers I have seen so far that remotely seem to address this question: in case of bigger T it is harder for the attacker to come up with the longest chain that will convince the network of his version of the history. But this argument does not appeal to me: if T is short, yes the attacker will come up with a given length of a chain faster than for a longer T, but it is also as easier for the rest of the network to grow the existing branch - so this does not seem to provide any benefits to the attacker.

Is there some minimal T that makes the network secure, and if there is, what does it depend upon at least qualitatively if no quantification is possible.



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