Hobbes' answer explains why live broadcast is currently not feasible from the Mars side. I'd like to complement it with why this is currently not feasible from the Earth side.
Mars power budgets are not generous, so by the time spacecraft transmissions get back to Earth they are incredibly faint. The only equipment used to reliably receive these signals is the Deep Space Network (or DSN), the largest and most sensitive scientific telecommunications system in the world.
Here's why this is a problem:
- The DSN has only 3 of the large 70m antennas (and you need the large ones for high data rates), which have to collectively cover all deep space missions (including Mars) for all countries which partner with NASA. Granted, each dish can be in contact with more than one mission at a time (there's an excellent page maintained by JPL indicating who is in active contact), but time is at a premium.
- Consequently, DSN time is very expensive. This question suggests costs of close to 5000 dollars an hour, plus setup and tear down fees.
- While there are 3 dishes, only one is going to be pointing at Mars at any given time, and there are a lot of Mars missions. Consequently slots are also at a premium. In fact, this issue is a real concern given the glut of Mars missions coming up in 2020.
Now, you might suggest "we shouldn't care how expensive or how exclusive it is, this is a flagship class mission" and these sorts of calls certainly do happen. The Mars Reconnaissance Orbiter commands 16 hours a day of DSN time. However, the Mars Reconnaissance Orbiter also states that this link tops out at 4 megabits per second (worst case is 500 kbps), which leads us to the nail in the coffin:
- Best-case data rates from Mars are on the order of 4 Mbps, whereas compressed-with-H.264 720p video needs 12 Mbps and 1080p needs closer to 22 Mbps. This rate from Mars is only achievable for a few months at closest approach. You would need to increase the link speed by a factor of at least 3 to get live, HD video even then.
Now, pre-recorded HD video is a different problem entirely, and is certainly feasible given enough local storage on the spacecraft and enough time to send it back to Earth bit-by-bit.
Edit -- As @Baldrickk points out in the comments:
The answer you link to is for very high quality video. Lossy video can
be a lot lower in bandwidth. from en.wikipedia.org/wiki/Bit_rate: 2.5
Mbit/s YouTube 720p videos (using H.264) 4.5 Mbit/s YouTube 1080p
videos (using H.264) so if you dropped the compression quality a
little more than YouTube uses, 4Mb/s would be enough for 1080p
content.
The question then becomes where OP wants to draw the line for "live HD television", but it's fair to say that on data rates alone, HD-ish video could be plausible.
That said, other obstacles on both the Mars and Earth side remain, and I don't believe the Mars 2020 mission will have live video.