MQA is kind of like a container that allows big audio files to be packed into a more manageable size and streamed to your home network. There are some other purported benefits when they are unpacked by an MQA compatible DAC, but I have not heard one to make a fair assessment.
That said, you can still play MQA files with any DAC, but your DAC just won't be able to take full advantage of all the tech when unpacking the MQA file. My current DAC is a McIntosh D100 and it is not an MQA DAC, but I am still able to stream MQA packed files and play them back, just not to their full advantage and I'm not sure how much (if anything) I'm missing, but they still sound really good to my ears!
A lot of DAC manufacturers are not on board with the whole MQA thing as I think it requires them to mold their DAC creations into something that may not be exactly what they intended their DACs to sound like. I think I read somewhere that there may also be some licensing issues, but not sure.
DSD is another of form of tech that was originally created by Sony Phillips for digitally recreating audible signals for their Super Audio CD format (SACD). Note the actual file extensions appear as DSF or DFF on your hard drive (not DSD). You can play those files through most DACs, but they need to be converted to PCM (the predecessor to DSD and more popular format of digital recreation) via playback software such as JRiver, Foobar, Roonm etc. rather than natively played back as DSD files in a DSD compatible DAC.
The DSD files that I have in my collection are some of the best sounding digital I have ever come across. Playing back DSD natively sounds excellent, but they still sound really good from DACs that can only handle PCM playback as well.
IMO a lot of that may be due to the care taken in the remastering of the analog master tapes rather than specifically being a DSD technology (more likely a combination of both). Most of the earlier DSD files out there are ripped from Super Audio CD's (SACD) and when Sony put those out they put a lot of care in transfer and remastering of those files ... so I think a big part of why they sound so good is that they were just handled with much more care than a typical PCM CD was. Just my theory of course, but I'm sticking to it!

BTW I have the Rega DAC too in my bedroom system.