This is what I would try to reach space: I would make a rope ladder system with poles extending at intervals to hold hydrogen balloons. I would space the balloons in a way that they won't affect other balloons or the ladder if they explode.
Also, there's a rather large difference between a multi-billion dollar internationally supported scientific operation, and a few million dollar marketing stunt.
If you can make a few hundred meters long carbon nano tube (which we can't right now), what's stopping you from making a balloon with a circumference of a few km, and how much cable could you lift up how high with that?
Since gravity decreases quadratically, the first miles are the ones that count.
If you're trying to get to space quickly, then much of the air-resistance losses are in the first 10 km.
Look up "rockoons" for more info on replacing a rocket's first stage with a balloon platform.