  The first named author introduced the notion of upper stability for metric
spaces as a relaxation of stability. The motivation was a search for a new
invariant to distinguish the class of reflexive Banach spaces from stable
metric spaces in the coarse and uniform category. In this paper we show that
property $Q$ does in fact imply upper stability. We also provide a direct proof
of the fact that reflexive spaces are upper stable by relating the latter
notion to the asymptotic structure of Banach spaces.
