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Let $B$ be a bifractional Brownian motion with parameters $H\in (0, 1)$ and$K\in(0,1]$. For any $n\geq1$, set $Z_n=\sum_{i=0}^{n-1}\big[n^{2HK}(B_{(i+1)/n}-B_{i/n})^2-\E((B_{i+1}-B_{i})^2)\big]$.We use the Malliavin calculus and the so-called Stein's method on Wiener chaosintroduced by Nourdin and Peccati \cite{NP09} to derive, in the case when$0<HK\leq3/4$, Berry-Ess\'een-type bounds for the Kolmogorov distance betweenthe law of the correct renormalization $V_n$ of $Z_n$ and the standard normallaw. Finally, we study almost sure central limit theorems for the sequence$V_n$.

Berry-Ess\'een bounds and almost sure CLT for the quadratic variation of the bifractional Brownian motion