Quick Answer: Yes, Liquid Notes supports Ableton Live. However, there are limitations in Live when it comes to exporting MIDI.
Update (Feb 2014): There is also a MIDI effect of Liquid Notes available now that plugs directly into Ableton Live 9 for a better workflow. It requires Max for Live.
Liquid Notes and Ableton LiveLiquid Notes does support Ableton Live (as any other sequencer or notation software), i.e. it is possible to route multitrack arrangements from Liquid Notes to Live. Sadly, Live doesn't support multitrack MIDI export - which is what Liquid Notes has been designed for - and also doesn't include tempo information in MIDI files when exporting your track.
We are aware of these problems and the limitations these provide to users of Ableton Live: consequently, we have already included a solution for the multi-channel MIDI export. Another fix for the missing tempo information in MIDI files is in the works already. Read more about it below.
Multi-channel MIDI export in LiveAbleton Live does not support multi-channel MIDI export. It is only possible to export single track files. With version 1.5 of Liquid Notes we have included a feature that allows you to combine single track files to multi-track files directly in Liquid Notes.
Watch the video with explanations on how to use this feature:
This feature is not restricted to Ableton Live only, you can use it with any other sequencer or in stand-alone mode whenever you want to import multiple single-track files and combine them to one arrangement.
Missing tempo information in MIDI export in LiveAbleton Live does not include tempo information in MIDI files when exporting your session. I.e. when opening the file in Liquid Notes it will play at the default value set in the MIDI file (i.e. 120bpm), which can be a far way off from the beats you set your track to in Live.
We have implemented a tempo adjustment feature in Liquid Notes 1.5.2, that allows you to adjust the tempo for any MIDI file you open. Find out more about it in this article - http://re-compose.desk.com/customer/portal/articles/1399080-how-do-i-adjust-the-playback-tempo-in-liquid-notes-.