What is the correct version please?
- The human brain capacity to do something.
- The human brain's capacity to do something.
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I would define "human brain capacity" differently than "human brain's capacity". It's partly emphasis — the former emphasizes human, while the latter puts the stress on brain —; but it's also that brain capacity is a quantity, while brain's capacity is a capability.
Thus, it makes sense to talk about The human brain's capacity to do something, but not really about The human brain capacity to do something.
You can say this several ways.
The capacity of the human brain — in other words, human brain capacity — is something to behold.
A human's brain capacity is something to behold.
The human brain's capacity for knowledge and adaptation is seemingly limitless.
All are correct.
The human brain's capacity to do something is amazing.
The capacity belongs to the brain, so we use the possessive form, which is
Note that the two examples you give are sentence fragments, since they do not have a finite verb.
To enlarge on Martha's answer, I think the quality/quantity distinction is the key to the answer, when you realise that "capacity" as a quality always belongs to somebody or something - "my capacity to ...", "the government's capacity ... ", "the capacity of the average citizen to ... ".
The quantity does not need to belong to something grammatically, (even though it will actually be the capacity of something): "The capacity is 400ml" is fine, whereas "The capacity to eat" is odd (not impossible; but the possessor of the capacity must have been established first).
Hence "the human brain's capacity" is possessed, and therefore can mean the quality or the quantity, whereas "human brain capacity" is not explicitly possessed - it is just a kind of capacity - and so can usually mean only the quantity.
The human brain's capacity is 600 billion neurons.