Yes, they'd be bound to set up live broadband streaming to Mars first. We could do that already, and maybe will do in the near future. We could easily have live streaming from our robots on Mars, if we were to set up a system or satellites orbiting Mars dedicated to communications to Earth - maybe using laser technology. And build more radio dishes or laser communications receivers, to receive the communications here back on Earth. It would cost a few billions I imagine but a tiny fraction of the total cost of a human mission.
I think that we should do that anyway if really keen on exploring Mars. It would make a huge difference to robotic exploration also.
Because at present the way they work is to get the data one day, and then use it to plan out operations for the next day because the bandwidth is so low, and so many spacecraft using it (often relayed through satellites that need to communiate their own observations as well) and there's the bandwidth for receiving it also - that you can't hope to communicate with your rovers more often.
So - at present, Curiosity could be as far away as Pluto and it would make almost no difference at all to the science team controlling it.
If they could literally control it with a response turn around of as little as 8 minutes at closest approach, and combined with some ideas used for online game streaming - then we could probably explore Mars almost as easily as we can explore e.g. the deep ocean bed here on Earth, with robots. And with real time live streaming in HD of everything they see.
As you will find from my other posts, I don't actually think humans are going to go down to the surface of Mars myself in the near future, say by the 2030s unless we can find a way to do that without introducing Earth microbes to Mars. It's too soon, we don't know enough about Mars yet or what effect Earth life could have on it (especially after a crash landing of a human occupied spacecraft).
So, more likely, it would be live streaming of a human mission to Mars orbit or to its moons Deimos and Phobos and then live streaming of all the movements of the robotic avatars they are controlling on the surface as they explore Mars, and experts on Earth would be able to look through the footage to see if you spot things that the explorers in Mars orbit missed in their telepresence exploration of Mars, and could advise them to have a second look at anything interesting.
And I think that we are bound to have humans exploring the Moon in more detail long before any mission to Mars. And those also of course live streamed - just depends on having the budget to implement it there as we already have the capability clearly from Apollo, much advanced now. Many exciting things to be found there I'm sure. If not NASA then for sure someone else will go back there before long. Either humans on the surface or robots with live streaming back to Earth.
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From the point of view of the individual photons of the radio waves, no time has elapsed at all on the passage from Mars to Earth. So I think it is reasonable to call it a live stream. For that matter we also call satellite broadcasts live streaming although there's a delay. A delayed live stream.
But as Benaya Christo says in his comment, not an interactive stream. Mission control won't be able to talk to them in real time from Earth. E.g. if an Apollo 13 moment arises and they say "Houston we have a problem" then they won't get any response at all for anything between 8.6 minutes and 42 minutes, and they are on their own to try to figure out a solution. How long does it take for a radio signal to go from Earth to Mars?
Many movies about human missions to Mars get this wrong. Short of hyperwave transmission through hyperspace, or tachyon transmissions if such ideas even make sense, we won't be able to do that.
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With the rovers, then I need to expand that a bit more. First there's a big difference between a delay of 8.6 minutes and a delay of one day. So broadband communications would make a huge difference. 167 two way interactions per day when closest instead of just 1. So probably more than a hundred times as much interaction - that by itself is a big thing.
But then there's artificial real time, which is based on building up a local model of the terrain and you then drive around it with any regions of uncertainty shown - so after driving around it a bit you soon have a detailed model of all the terrain and no uncertainty for a particular region the rover is exploring, then you can explore it pretty much in real time once you have it reasonably well characterized.
Add in a bit of autonomy and self righting rovers, and no reason why we can't be driving around on the surface at close to the speed the astronauts drove their Moon buggy on the moon. Do Curiosity's entire journey so far in the first day of exploration most likely. And in the meantime you've developed a detailed terrain map, and started to map out pretty much all the individual boulders also - so before long you are driving up and down Mount Sharp wherever you want to go using artificial real time at close to the same speed you could do with a telerobotic rover on the Moon, using images you have already combined with other images from other directions on previous passes to have a detailed 3D "game environment" to drive in.
So - I see humans as most likely being useful in orbit around Mars or in settlement on Deimos, so that they can control rovers anywhere on Mars, which they'll be able to do with no risk of contaminating it with Earth life.
With a spectacular view of the planet, which looks much more home like from above, no dust storms also to block out your view, no 7 minutes of terror for landing, 24/7 communication with Earth, and controlling rovers anywhere on the surface, places far too steep or dangerous for humans, underground caves, mini planes flying up and down the Valles Marineres, robotic rovers exploring the Martian geysers close up and watching them erupt, ...
I expect that's what we'll have televised if humans do end up going to Mars (orbit) from Earth. With everything they see on the surface of Mars also live streamed to Earth And what an exciting mission it would be to follow from Earth.