A good transient response, good bandwidth (flat to 40Hz at least), excellent timbral quality, pleasing or at least unobtrusive looks, seamless crossover, freedom from sibilance, and good dynamics.
If I'm spending more than £3000, and it certainly looks like I will have to if I'm hoping to get any upgrade from my Tannoy Berkeley's, then an exemplary spec sheet is also a must (esp dispersion/radiation patterns).
Loudspeaker upgrades are notoriously difficult affairs. It would be very easy to go around in circles I'm sure.
If I had to chose only one quality, then it would have to be that they don't annoy.
Many loudspeakers I have heard do.