Too many monics

Since both $P(Q(x))$ and $Q(P(x))$ have four distinct real zeros, both $P(x)$ and $Q(x)$ must have two distinct real zeros. Hence, we can write $P(x) = (x - h_1)^2 - k_{1}^2$ and $Q(x) = (x - h_{2})^2 - k_{2}^2$ .

The zeros of $P(x)$ are $h_{1} \pm k_{1}$ , so the zeros of $P(Q(x))$ are the roots of $Q(x) = h_{1} \pm k_{1}$ . Since the function $Q(x) = (x - h_{2})^2 - k_{2}^2$ is symmetric around $x = h_{2}$ , the roots of $Q(x) = h_{1} + k_{1}$ must be symmetric around $x = h_{2}$ , and the roots of $Q(x) = h_{1} - k_{1}$ must also be symmetric around $x = h_{2}$ . The values $-23, -21, -17$ , and $-15$ are symmetric around $-19$ , so $h_{2} = -19$ . Then $Q(x) = (x + 19)^2 - k_{2}^2$ .

Since $Q(x)$ is monic, $Q(x)$ is decreasing on the interval $(-\infty,-19]$ and increasing on the interval $[-19,\infty)$ . Therefore, the roots of $Q(x) = h_{1} - k_{1}$ are $-21$ and $-17$ , and the roots of $Q(x) = h_{1} + k_{1}$ are $-23$ and $-15$ . Hence, $Q(-17) = h_{1} - k_{1}$ and $Q(-15) = h_{1} + k_{1}$ , so $Q(-15) - Q(-17) = 2k_{1}$ . But $Q(-15) - Q(-17) = [(-15 + 19)^2 - k_{2}^2]- [(-17 + 19)^2 - k_{2}^2] = 16 - 4 = 12$ , so $2k_{1} = 12$ , which means $k_{1} = 6$ .

Similarly, the zeros of $Q(x)$ are $h_2 \pm k_{2}$ , so the zeros of $Q(P(x))$ are the roots of $P(x) = h_{2} \pm k_{2}$ . Since the function $P(x) = (x - h_{1})^2 - k_{1}^2$ is symmetric around $x = h_{1}$ , the roots of $P(x) = h_{2} + k_{2}$ must be symmetric around $x = h_{1}$ , and the roots of $P(x) = h_{2} - k_{2}$ must also be symmetric around $x = h_{1}$ . The values $-59, -57, -51,$ and $-49$ are symmetric around $-54$ , so $h_{1} = -54$ . Then $P(x) = (x + 54)^2 - k_{1}^2$ .

Since $P(x)$ is monic, $P(x)$ is decreasing on the interval $(-\infty,-54]$ and increasing on the interval $[-54,\infty)$ . Therefore, the roots of $P(x) = h_{2} - k_{2}$ are $-57$ and $-51$ , and the roots of $P(x) = h_{2} + k_{2}$ are $-59$ and $-49$ . Hence, $P(-51) = h_{2} - k_{2}$ and $P(-49) = h_{2} + k_{2}$ , so $P(-49) - P(-51) = 2k_{2}$ . But $P(-49) - P(-51) = [(-49 + 54)^2 - k_1^2] - [(-51 + 54)^2 - k_1^2] = 25 - 9 = 16$ , so $2k_{2} = 16$ , which means $k_{2} = 8$ .

The minimum value of $P(x) = (x - h_{1})^2 - k_{1}^2$ is $-k_{1}^2$ , and the minimum value of $Q(x) = (x - h_{2})^2 - k_{2}^2$ is $-k_{2}^2$ , so the sum of their minimum values is $-k_{1}^2 - k_{2}^2 = -36 - 64 = \boxed{-100}$ .