Efficiency of speakers?

What do you mean "faster?"

Quicker transient response Pawlowski.

Meaning the driver can push and recover quicker, and without fatigue. More deep level signal can be heard that way.

There is not a direct connection between speaker sensitivity and transient response. That said, some speakers acheive high sensitivity by using massive magnet structures coupled with very light weight cones which should make them very quick sounding drivers. But again there is no direct connection the way you're implying.

I don't think that your question can be answered with the single variable of amp wattage. The speaker efficiency that you are referring to is a measure of sound pressure level. In other words, what is the output level of a speaker driven by 1 watt measured at a distance of 1 meter on axis. The transient response, or "quickness", of a speaker is something different. Bigger amps in terms of wattage will not necessarily give better transients. They may or they may not. For example, for dynamic speakers, cone movement must be well controlled or this will blur transients. If the larger wattage amp has better control of the cone, it will produce better transisents. If not, it won't. It depends on the amp's design, of which wattage is only one part. The speaker itself is also a variable. Electrostats, for example, are lighter than cones and typically have good transients because the lighter driver is easier to start and stop, even if the cone and electrostat have the same "efficiency", all other things equal of course. So I think that the answer to your question is "maybe". It depends on a bunch of other variables besides the sound pressure level measurement of a speaker.