In this paper, it is shown that the Berezin–Toeplitz operator Tg is compact or in the Schatten class Sp of the Segal–Bargmann space for 1⩽p<∞ whenever g˜(s)∈C0(Cn) (vanishes at infinity) or g˜(s)∈Lp(Cn,dv), respectively, for some s with 0<s<14, where g˜(s) is the heat transform of g on Cn. Moreover, we show that compactness of Tg implies that g˜(s) is in C0(Cn) for all s>14 and use this to show that, for g∈BMO1(Cn), we have g˜(s) is in C0(Cn) for some s>0 only if g˜(s) is in C0(Cn) for all s>0. This “backwards heat flow” result seems to be unknown for g∈BMO1 and even g∈L∞. Finally, we show that our compactness and vanishing “backwards heat flow” results hold in the context of the weighted Bergman space La2(Bn,dvα), where the “heat flow” g˜(s) is replaced by the Berezin transform Bα(g) on La2(Bn,dvα) for α>−1.