What is a convex multistage problem?

We define a convex multistage problem with $N$ stages as follows:

\min_{z_i}\ & \sum_{i=1}^N \frac{1}{2}z_i^T H_i z_i + f_i^T z_i \ \;\;\qquad\,\,\qquad\qquad\qquad\qquad \textsf{(separable objective)}\\
\text{s.t.}\ & \underline{z}_i \leq z_i \leq \bar{z}_i\enspace, \qquad\qquad\quad\ \, \, i=1,\dots,N\enspace,\quad\;\;\,\,\, \textsf{(bounds)}\\
& A_i z_i \leq b_i\enspace, \qquad\qquad\qquad\ \ \, i=1,\dots,N\enspace,\qquad \textsf{(affine inequalities)}\\
& z_i^T Q_{i,j} z_i + g_{i,j}^T z_i \leq r_{i,j}^2 \enspace,\quad i=1,\dots,N\enspace,\ \qquad \textsf{(quadratic inequalities)}\\
&\qquad\qquad\qquad\qquad\qquad\quad\ \ j=1,\dots,q_i\enspace,\\
& C_i z_i + D_{i+1} z_{i+1} = c_i\enspace, \quad \, \; \; i=1,\dots,N-1\ \quad \textsf{(equalities coupling 2 variables)}

with $H_i\succeq 0\ \forall i$, $Q_{i,j}\succ 0\ \forall i,j$ and where $z_i$ are the so-called stage variables. Note that these do not necessarily have to have the same dimension for each stage.