The original idea of QSS was to bolt an ad hoc step length control mechanism onto the most basic explicit method, forward Euler. So at its core, the original QSS method is exactly a traditional explicit integration, except that the length of any individual step is not adjusted based on feedback about how well the step actually did.

Why would you do that? The motivation was to enable parallel execution of large modularly-composed systems, without getting bogged down with iterations. Basically every important design decision in QSS methods comes directly from, or is at least deeply influenced by, the desire to avoid iterations between components in the simulation. They want to avoid large communication overheads, and they want to enable components to step forward in time as independently of each other as possible. The dream is to enable the simulation of large systems by just connecting components together, and letting each one broadcast what it plans to do over the next time step, and how long it will do it. Thus the core idea of QSS is to choose the step length a component can take before it takes the step, and to not get too inquisitive about how well the step actually did.

I always thought that QSS methods would benefit from adapting a trust region approach to at least control errors going forward in time, but as far as I know, they never did.

By the way, later versions of QSS introduced some implicit-flavored methods. I believe they are still only implicit in the local information (i.e., only state variables defined within a component get an implicit model), and that state variables defined in other, communicating, components still have an unchanging, forward-projected model. However, it's been a long time since I looked at those methods, and I might be wrong about that.