First, to be pedantic, we aren't descended from apes, just very closely related to them with a common ancestor.
Second, is there any evidence that the evolutionary history of intelligence was linear? How do we know that there wasn't a binary-like change where one generation didn't have our level of intelligence and the next did, and all humans today are descended from that single change, as opposed to it happening gradually?
If you are going to be pedantic, genetically we are not just descended from apes, we are apes. If you look at the evolutionary tree of the apes and monkeys, it makes no sense to consider humans outside the ape group. Chimpanzees and humans are more closely related than Chimpanzees and Gorillas for example.
The only reason we consider humans to be descended from rather than being apes, is that we don't look at the evolutionary tree from the outside. Biologically it makes no sense.
> How do we know that there wasn't a binary-like change where one generation didn't have our level of intelligence and the next did, and all humans today are descended from that single change,
This question seems kind of confused. But we can say with complete certainty that it's not the case that "one generation didn't have our level of intelligence and the next did" because generations don't happen at any particular time. Imagine your ancestor through the paternal line 30 generations back from you (that is, your father's father's father's... father). If you call him generation 1, then you're generation 31. But other people born in the same year as you, also descended from him, will not all be generation 31. And if you married one of them who was, for the sake of argument, only 28 generations removed from him, your children would be, in this labeling scheme, simultaneously generation 32 and generation 29.
The reason I say your question is confused is that the model where there was a "single change", and all humans today descend from it, requires several generations to take hold (as it gradually increases its population share), but you appear to posit only one.
The most simplistic scenario in which a change could happen in "one generation" is if there is one man today born with a mutated gene that protects him against X, and tomorrow X happens which kills all men in the world except him, leaving him to impregnate many women, and all future generations would be descended from him. Not saying that has ever happened (I've no idea, for all I know it might have but does seem fairly unlikely), but in theory it's possible.
The question is how you would get a single allele that gets you from lower apes to fully sapient humans in one mutation, in one individual, that also protects against something which killed off everyone else.
It wouldn't have to be a single mutation, but let's say intelligence is linear from 1 (apes) to 5 (humans). At some point an animal mutated from 1 to 2, then 2 to 3, etc. Each step the more intelligent version ended up killing off the weaker version (either through violence or just through better survival in nature), and we ended up where we are at 5, which all humans begin at and our surroundings (i.e. nurture not nature) can modify that 5 down to 4.9 or up to 5.1
I'm incredibly not an expert on evolution (it shows I'm sure!), but from a logical point of view that seems like it's feasible at least - whether it's incredibly likely or incredibly unlikely I've no idea.
How do we know that there wasn't a binary-like change where one generation didn't have our level of intelligence and the next did, and all humans today are descended from that single change, as opposed to it happening gradually?
Because it's very, very rare for evolution to actually work that way.
Second, is there any evidence that the evolutionary history of intelligence was linear? How do we know that there wasn't a binary-like change where one generation didn't have our level of intelligence and the next did, and all humans today are descended from that single change, as opposed to it happening gradually?