Livestreaming Youtube
What you are asking is extremely difficult without some very expensive and complex software and video studio equipment. I learned some techniques in After Effects that were supposed to make an enlarged video look better (certain blurs and sharpeners, deinterlacing, etc) but I was NEVER satisfied with the results. Every technique made the enlarged version blurry or grainy. These techniques I just mentioned do improve the quality somewhat, but does not magically or vastly improve the quality like what everybody expects expects (not saying you do but in general).

It's would be easier for you just to stream from Youtube Live since it can be set to automatically record and upload your stream once its completed. It does so in HD or at least matches the quality of the stream.

I also find Youtubes general stream quality to be much better than LiveStream's.

Here is the official Youtube user guide on how to setup live streams. The guide little lengthy but it explains all the steps quite thoroughly.

http://support.google.com/youtube/bin/static.py?hl=en&amp;page=guide.cs&amp;guide=2474025

Notice Step 1, Part 8 with information about recording the live stream. Just remember that if your live stream is going to go over 4 hours, which in my opinion is tremendously long, you cannot use the recording feature. If this is one of the rare events or streams that would go over that time limit (maybe a huge concert or music festival) then a solution would be to have an "intermission" where you stop streaming at 3 hours and 50 minutes just to be safe. Then have a nice 10 minute break to do whatever you want!

One important thing to note when recording live streams with Youtube is that you can't edit the recorded stream once it exceeds two hours. This is kind of lame but I understand why it was necessary.

Additionally, it would be a good idea to record a local copy using an HD screen and audio recorder. In this answer I listed a few popular examples of screen recording software.