I'm still a little baffled about the archives, but the player ended up choosing another improvement so I didn't have to deal with it (but it's probably coming one of these days).
Let me take another stab at it.
So, the quarantine playbook says:
Archives: stasis includes a workspace in the form of its records and historical archives. Access them and you can get to the bottom of the past like a savvyhead getting to the bottom of some tech shit (cf).
A "workspace" is a specific kind of resource in AW. The "cf" means, look up how it works by consulting the savvyhead's workspace rules. Those say:
When you go into your workspace and dedicate yourself to making a thing, or to getting to the bottom of some shit, decide what and tell the MC. The MC will tell you "sure, no problem, but..." and then 1 to 4 of the following:
... followed by the list of options.
"Getting to the bottom of shit" is a technical term in AW, believe it or not. Getting to the bottom of shit is one of the things you can do in a workspace. The idea is, you can gain knowledge, like to solve a mystery or something. Like, if a savvyhead has a piece of tech, and he wants to figure out what it does, or how to fix it, or whatever else, he can "get to the bottom" of that shit.
The quarantine can't do that with tech. But he can "get to the bottom of the past" -- he can go in with the intention of learning something specific about the way the world was in olden times, and then the MC can choose conditions from the workspace rules that have to be met before the she's obliged to give answers.
For example, maybe in your apocalypse, there was a plague that wiped everyone out. Let's say the quarantine suspects it might have been a biological weapon, but wants to prove it. He goes into the archive looking to get to the bottom of that shit. The MC says, "Sure, no problem, but it's going to take months of research to make sense of all of the epidemiological data. Or, if you get your CO's clearance to access the classified stuff, you could probably get a definitive answer in a few days of work. Too bad he's still frozen."
The answers aren't free, right? The quarantine has to meet conditions to get them, and it's the MC's job to make those conditions not boring, and make the answers seem real. That's one difference between the archives and the memory move: for the memory move, the answers are free, no conditions. Also, a lot of the memory move questions pertain to personal matters, or matters of opinion (eg. "Could we have stopped it?"), stuff that might not be recorded in an archive.
Also, the quarantine can "get to the bottom of the past" -- he can learn about the way things were. That doesn't automatically mean he can apply that knowledge to make the present like the past. I mean, the archive might tell me how they built wind turbines 75 years earlier, when there was a power grid, and mills producing high quality steel alloys, and factories assembling high efficiency electrical components, and you could hire a dozen guys to erect the thing. Having that information doesn't mean it's now easy to build such a thing in Apocalypse World. But maybe if I can pass that info to Bran the savvyhead, he can go into his own workspace and see what he can do to adapt the information. And then maybe some of his conditions are, you need a part from another hardhold, you'll need a truck to haul it back, and you need the volunteer labour of the chopper's gang, etc. Moves should snowball, right? And then, at the end, you get a very small piece of the past, which the MC is permitted to look at through crosshairs.
Does that make sense?