How many negative eigenvalues can $AB + BA$ have when $A$ and $B$ are symmetric positive definite? Let $A, B \in \mathbb R^{n\times n}$ symmetric positive definite. Clearly, any eigenvector $v$ of either $A$ or $B$ is such that (taking for example a nonzero eigenvector of $A$ with eigenvalue $\lambda$)
$$
v^T(AB + BA) v = 2 \lambda (v^TBv) > 0,
$$
so $AB + BA$ has at least one positive eigenvalue. From this answer, we conclude that, in dimension 2, $AB + BA$ has at most one negative eigenvalue. 
What is the situation when $n$ is general? Can we have more than $\lfloor n/2\rfloor$ negative eigenvalues? (the value $\lfloor n/2\rfloor$ can be obtained by taking block diagonal matrices with 2x2 blocks corresponding to the 2-dimensional example.)
 A: $AB+BA$ can have $n-1$ negative eigenvalues (it has at least one positive eigenvalue because its trace is positive). Let $B$ be the symmetric Toeplitz matrix whose first row is $(1,t,t^3,t^5,\ldots,t^{2n-3})$:
$$
B=\pmatrix{
1&t&t^3&\cdots&t^{2n-3}\\
t&\ddots&\ddots&\ddots&\vdots\\
t^3&\ddots&\ddots&\ddots&\vdots\\
\vdots&\ddots&\ddots&\ddots&t^3\\
t^{2n-3}&\cdots&t^3&t&1}.
$$
When $t>0$ is sufficiently small, $B$ is positive definite. Let $A^{1/2}=\operatorname{diag}(1,t^2,t^4,\ldots,t^{2n-2})$. Then
$$
A^{1/2}BA^{-1/2}=\pmatrix{
1&t^{-1}&t^{-1}&\cdots&t^{-1}\\
t^3&1&t^{-1}&\cdots&t^{-1}\\
t^7&t^3&\ddots&\ddots&\vdots\\
\vdots&\ddots&\ddots&1&t^{-1}\\
t^{4n-5}&\cdots&t^7&t^3&1}.
$$
Now put $S_t=t\,(A^{1/2}BA^{-1/2}+A^{-1/2}BA^{1/2})$. Then
$$
S=\lim_{t\to0}S_t
=\pmatrix{0&1&\cdots&1\\ 1&\ddots&\ddots&\vdots\\ \vdots&\ddots&\ddots&1\\ 1&\cdots&1&0}
$$
has an eigenvalue $-1$ of multiplicity $n-1$. Therefore $S_t$ has $n-1$ negative eigenvalues when $t$ is small. Consequently, by Sylvester's law of inertia,
$$
AB+BA=\frac{1}{t}A^{1/2}S_tA^{1/2}
$$
has $n-1$ negative eigenvalues as well when $t>0$ is sufficiently small.

Edit. I have just discovered that a different example pair of $A$ and $B$ based on recursive construction had already been given in an answer to another question.
