  We study ancient mean curvature flows in $\mathbb{R}^{n+1}$ whose tangent
flow at $-\infty$ is a shrinking cylinder $\mathbb{R}^{k}\times
S^{n-k}(\sqrt{2(n-k)|t|})$, where $1\leq k\leq n-1$. We prove that the
cylindrical profile function $u$ of these flows have the asymptotics
$u(y,\omega,\tau)= (y^\top Qy -2\textrm{tr}(Q))/|\tau| + o(|\tau|^{-1})$ as
$\tau\to -\infty$, where the cylindrical matrix $Q$ is a constant symmetric
$k\times k$ matrix whose eigenvalues are quantized to be either 0 or
$-\frac{\sqrt{2(n-k)}}{4}$. Compared with the bubble-sheet quantization theorem
in $\mathbb{R}^{4}$ obtained by Haslhofer and the first author, this theorem
has full generality in the sense of removing noncollapsing condition and being
valid for all dimensions. In addition, we establish symmetry improvement
theorem which generalizes the corresponding results of Brendle-Choi and the
second author to all dimensions. Finally, we give some geometric applications
of the two theorems. In particular, we obtain the asymptotics, compactness and
$\textrm{O}(n-k+1)$ symmetry of $k$-ovals in $\mathbb{R}^{n+1}$ which are
ancient noncollapsed flows in $\mathbb{R}^{n+1}$ satisfying full rank condition
that $\textrm{rk}(Q)=k$, and we also obtain the classification of ancient
noncollapsed flows in $\mathbb{R}^{n+1}$ satisfying vanishing rank condition
that $\textrm{rk}(Q)=0$.
