This will be in no particular order, and I'll add to this as I go along. Comments are welcome.
The Tao of History
Aaaand of course this is the one my brain decides to start with. I guess it makes sense; they do have a clock as their emblem.
The Tao are all about recreating traditions and staying true to the past. Since research methods improve over time, Chronotech allows them to send back information about the past, from a time in the future when they have a better understanding of it. Their reenactments become more accurate.
There's also the potential of "reenacting" events that have yet to actually happen. This is a bit of a waste of temporal bandwidth, since information from the future is much more limited than information from the past. However, if the Tao spend their bandwidth on this, it means other people can spend theirs on other stuff, and that can rack up some serious money for the Tao. They could really become future-information brokers, charging huge amounts for their performances.
Likely Chronotech score: 4, with 4 for citizens as well.
Mechanica
Wiring future-information directly into your brain is a no-no in Mechanica. That's changing the brain, and changing the brain is not part of their definition of Humanity.
Mechanica does care about money and elections. As a plutocracy, those are basically the same thing for them. Here's where things become a serious pain in the butt for Mechanica. Knowing the future means they know who's going to win the election. However, because interaction permits alteration, the results of the election are actually still in flux. This is a more complex game than Mechanica can actually handle. With their low Metatech and Cognitech scores, they're unlikely to be able to capitalize on the information from Chronotech very effectively, and their leadership is going to fluctuate more heavily as a result. They're probably going to be more vulnerable to outside manipulation.
Being skilled at Stringtech means that Mechanica is likely to have a fair number of physical time machines sitting around. This will give them a much-needed boost when it comes to research. The bandwidth on a physical time machine is much larger than the bandwidth on ordinary Chronotech, and if Mechanica can defend them properly and limit access, they should reach quite a distance into the future.
Likely Chronotech score: 3 with 2 for the citizenry, but screw it we'll use our Competitive Advantage in Stringtech to do that instead.
The Logicians
As believers in (and enforcers of) strict hierarchy, it's likely that the leadership of the Logicians will have access to Chronotech and the rank-and-file will not. This makes them easier to predict and easier to keep in line, which means that the Logician monarchy won't have to worry about the observer effect wrecking their predictions.
Of course, the Logicians will really want to know how the future goes for them. They really care about having humanity make it as far into the future as possible; everything they do has that as a long-term goal. They're likely to be on the lookout for longer- and longer-term chronal channels, without worrying too much about the bandwidth of those channels. I can imagine them creating a huge concordance of predictions and waiting to hear the few bits of data that will shape their future.
Interestingly, the Logicians are also one of the few groups that would totally lie to themselves to get a better and more efficient future… so as long as their descendants give them actionable information that puts them in a better situation, it doesn't even need to be correct.
Likely Chronotech score: 3, or (null) for the rank-and-file.
