The phenomena of ducted fan assemblies inducing pitching moments whilst in transition from vertical to horizontal flight paths is a problem I have seen mentioned by many designers and engineers of VTOL aircraft. I am wondering if the quadratic arrangement(a fan placed at each of the outermost corners of the aircraft) would help alleviate these issues. It would seem to make sense as the front and rear fan pitching moments about a central CG would cancel each other out, though still causing stresses on the airframe. Are these pitching moments still a problem in quad fan configurations, and has any research been done into the inlets of ducted fans to stop these pitching moments?
The Doak VZ-4 twin ducted propulsor flew in 1958, the Bell X-22 took that up to four in 1966. So yes, people have been thinking about it for over half a century.
Analogously to any tailless aircraft, the side-by-side twin is sensitive to CG in vertical mode and trim must be closely maintained. Although not ducted, the Boeing V-22 Osprey convertiplane suffers from this issue. It is not often an operational limitation, but it does need crew/staff training and careful management.
Similarly the quad arrangement brings the same benefits as the tandem wing in allowing a wide range of CG positions. However the quadrotor does not scale well to full-size aircraft, as discussed at Why haven't quadcopters been scaled up yet?