Simple question but longish answer, so my apologies in advance.
QS is not the same as UHJ although the matrixing has similarities (e.g.. the phase shifters). UHJ, like QS, is encoded to two channels but thereafter the similarities end, really. QS is encoded from and decoded to so-called discreet speaker feeds whilst in ambisonic (UHJ and B-format) the speaker outputs combine around one's head to reconstruct the original soundfield as was around the ambisonic microphone. You can try putting UHJ encoded material through a QS decoder to find how it sounds to you, but it will not be as intended. Likewise putting QS through an ambisonic decoder may give pleasing results, though not as intended. Nimbus Records have produced some G-format (not a typo) material from the UHJ masters that are intended for replay using a DVD player with DTS & MLP (4.0) playback (into a virtual quad array, basically). This format is a pre decoded form for people without an ambisonic decoder. One such release is A Midsummer Night's Dream (NI9009).
I have the original Mike Oldfield Boxed quad LPs, which are QS encoded and sound pretty well using ambisonic playback, but I have never tried it over a quad (QS) system. But this is the only quad stuff I have.
In ambisonics, the soundfield is 'measured' by the mic and 'decomposed' (see Spherical Harmonic Decomposition) into a pressure (omni) signal (W) and three, orthogonal vector signals (XYZ), which are not loudspeaker feeds. This soundfield scene description is used for reconstruction of the soundfield by recombining the 3 (for horizontal, WXY) or four (for spherical, WXYZ) signals appropriate for your particular loudspeaker layout; ambisonics is speaker number and position agnostic and thus you tell the decoder how many speakers you are using and where the speakers are placed. The minimum number of speakers for horizontal reconstruction is four, which works well for smallish arrays. For larger arrays, or where the front-back ratio to left-right ratio is large, 6 speakers may give better stability for side source images. In principle, you can use as many more speakers as you like, although there is next to nothing to be gained from going to more that 8 (horizontal) for a first order (aka. FOA) signal set. Besides, I know of no domestic ambi decoders that cater for more than 7 speakers (inc a CF). The signal feed to each speaker is adjusted according to the number and placement of the speakers. For first order playback in a small space, such as my sitting room, the decoders are dual band using velocity optimised decoding (rV) below 200 Hz and energy optimised (rE) above 700 Hz, with the perceptual cues from each band kept in step to avoid them conflicting (phase and energy cues).
I hope this helps more than it confuses.