Generally there are two ways of how an arpeggiator syncs to MIDI clock:
The arpeggiator starts immediately when the first key is pressed and uses the tempo of MIDI clock. This means, if I press the key a little too early or too late, this mistake will be continued and all generated notes will be off beat. My synthesizer works like this, and I hate it.
The arpeggiator starts exactly on a beat, for example a 16th, in sync to MIDI clock. This means, if I press the key a little too late, the first note will be dropped. If I press the key a little too early (less than a 16th), the first generated note will be perfectly in time, and all further notes as well. The arpeggiator of Ableton Live works like this. But it’s not easy to press keys just this little bit too early.
What about a new method which tolerates human imperfection - I haven’t found this anywhere so far. It means that the arpeggiator starts immediately when a key is pressed, but reduces the mistake with every step, so after a few steps the timing is perfectly in sync to MIDI clock. This is the way a human player would correct his / her own imperfect timing. I could imagine this as a real advantage for every live-musician who wants to use the arpeggiator.