Since (k+1>0) for all (k\ge 1), each term

$$
(k+1)z_k^2
$$

is non-negative. Therefore,

$$
\sum_{k=1}^{n}(k+1)z_k^2 \ge 0.
$$

To determine whether this lower bound is attainable, consider

$$
z_1=z_2=\cdots=z_n=0.
$$

Then the constraint

$$
\sum_{k=1}^{n} z_k = 0
$$

is satisfied.

Substituting these values into the objective function gives

$$
\sum_{k=1}^{n}(k+1)z_k^2 = 0
$$

Hence the minimum value is

$$
\boxed{0}.
$$

The minimum is attained when

$$
z_1=z_2=\cdots=z_n=0.
$$