I think it’s a mistake to view the Hard Problem as unique to materialism. Idealism can’t answer it either, and generally doesn’t try to. IMO, the problem is not really about matter, but about description or explanation.
(I also don’t think it’s unique to “formal” systems or approaches, except in a sense so broad that any philosophy that could ever be done is “formal,” because it involves strings of words and/or arguments.)
The Hard Problem is very similar to the problem of existence – “why is there something rather than nothing?” Both of these are questions about what “animates” or “turns on” any given description – what makes a description (such as a formal system) more than “mere words on a page.” This is a distinctive class of problem because any familiar kind of explanation would simply become part of the description, and thereby be subject to the exact same problem.
If you add some sort of “existence-maker” mechanism to your description of what exists, you’re still open to the objection that the entire description, existence-maker and all, could just as well be an inert logical structure, without the extra magic of existence. This is a pretty familiar, standard point in the context of the existence question, but in discussions about consciousness, the analogous point tends to get buried under arguments about whether or not there is more of a problem for certain kinds of description – “material” or “formal” or “functional” ones, or whatever.
It seems to me that this is a problem for descriptions, period. If you look at the various dualistic and idealistic systems that have been proposed, they tend to be, well … systems: descriptive accounts of what is supposed to exist (some or all of it mental/spiritual), along with some arguments about why we should assent to the description, but nothing inherent in them to light the flame and turn these descriptions necessarily into the realities they talk about. These systems do claim that the flame is in fact lit, but they generally treat this as self-evident via Descartes’ cogito or similar. At least one mind/spirit exists (by cogito), and here are some things it can conclude a priori about other existents – Leibniz’s various principles, McTaggart’s theory of determining correspondences, or whatever – and we’re off to the races.
These can be perfectly fine theories of what mind/spirit is, insofar as it exists, but they simply do not touch why/how it exists: you need the spark of a cogito to get things started, and the cogito doesn’t leave you any less in the dark about why there’s an existing mind (instead of there not being one). It just convinces you that there is one. And once you’ve decided to work within a frame where that is taken as given, you’ve given up on Hard Problems. These theories only “explain” the ineluctable experiencey-ness of experience in the way that the observation “as a matter of fact, something exists” explains why there is something rather than nothing – which is to say, not at all.
It seems intuitively clear to me that these Hard Problems are unanswerable, because they ask for something that is incompatible with what we take to constitute an “answer” to a question. They ask for an argument that some description is necessarily animated, that there’s no mystery about how it becomes more than words on a page because there is something impossible about the merely-words version of it. But such an argument is either:
(1) An argument for purely logical necessity, i.e. necessity within the terms of the description, in which case the necessity property is just one more fact about the description and could be as “mere” as the rest, or
(2) An argument that the description gets necessarily lit up by the animating fire of something else that already has it, in which case we need some initial spark to start things up, one that is not explained within the terms of the description. Generally this spark is supposed to be “obvious” / a priori, but the fact that we have a priori knowledge of something doesn’t constitute an explanation of why we have that knowledge, so this doesn’t get the job done.