High damping factor, ample headroom, and speakers designed with a Kick-bass preference (does not necessarily need to be called a Kick-bass speaker).
For example, the Rainbow SLC 165 Kick's (woofer) I have, have 1/3rd the xmax of the "classic" version - the Rainbow SLC 165 (sorry I don't remember the specific mm's). The Kick version must (according to Rainbow) be used with an 80hz highpass. They are specifically designed to offer a lot of punch but don't play as low as a more classical high-exursion speaker.
So I guess that attack would also be a feature of the technical specifications of a speaker (qts, qes, fs, vas etc) but I do not know to be honest
This is why in some cars the mid-bass can be very smooth and not so much muffled as delicate yet firm, whilst in my car the mid-bass is agressive without being overpowering - some judges prefer a strong mid-bass and others feel overpowered by it.
With a good set of classic speakers you can get away with very little in the way of subwoofers, Kick-design speakers require a subwoofer to perform decently.
And obviously having a high damping factor to control cone movement is essential, as is ample headroom to deal with the transient peaks in music.