Likes, retweets and quote tweets each surface a different kind of tweet. They are not interchangeable — which is why the anthology keeps a separate top 5,000 for each, and lets you order by any of the three.
Likes
Tweets people wanted to see more of. A like is a direct input to each user's algorithmic feed. Liking is cheap, private-feeling, and frequent — so the likes ranking is the closest thing to "what the community enjoyed reading".
Retweets
Tweets worth signal-boosting. A retweet is an explicit, public endorsement to one's own followers — costlier than a like, and so a stronger claim: this deserves a bigger audience.
Quote tweets
Tweets people wanted to respond to or build on. The noisiest of the three: it captures generativity, but also disagreement, as well as tweets that explicitly ask to be quoted. One important difference from the other two: this is not Twitter's global quote count — it counts only quote tweets written by other archive accounts. More below.
How the three relate
Empirically, likes and retweets correlate strongly across everything the 340 accounts wrote (Pearson 0.81); both are nearly uncorrelated with quote tweet count (0.10 and 0.09). The site's three top-5,000 lists reflect this: likes and retweets share about two-thirds of their entries, while the quote tweet list overlaps very little with either — it surfaces a distinctly different slice of the archive. Concretely: ~80% of tweets on the quote tweet list appear on neither of the other two.
Part of that decorrelation is mechanical rather than behavioural. Likes and retweets are global counts over all of Twitter; the quote tweet count is taken within the archive, where most tweets score zero or one — and counts that small are mostly noise to a correlation. But the difference is real too: the quote tweet list genuinely surfaces what resonated inside the community, rather than what travelled furthest outside it.
About the quote tweet metric in particular
Quote tweet counts here are corpus-internal: they count only quote tweets written by other archive accounts — a few hundred people, not all of Twitter. So the numbers are small (most tweets in the anthology have zero or one), and each is a lower bound on the tweet's true quote count.
What the count tracks is notable enough that someone in the community responded publicly. That overlaps substantially with worth including in an anthology of insight, but the two are not identical. Expect to find ideas generative enough to inspire responses, but also things that provoked enough disagreement to warrant a public rebuttal, and the occasional "quote this with your…" prompt.