Before the texture goes on the ceiling, you're supposed to mud and tape those joints so there aren't any gaps, and there shouldn't be that much of a gap to start with on drywall. Though perhaps this is something other than drywall.
To fill the gaps now, you've got several options, joint compound, spackling, caulk, or put back up some trim. However, without tape, joint compound is going to crack. Spackling is designed for much smaller gaps. Caulk will allow the two surfaces to move without cracking, but it will never have a smooth look over this large of a gap.
The only two "right" ways to fix this that I can see are to put back up some trim/crown molding, or to knock off the texture on the ceiling so you can fill with joint compound and tape it. If you do the latter, you'd need to knock off a 12" span of the texture, and it's difficult to replace that with something that looks the same.