Border rank of powers of ternary quadratic forms

We determine the border rank of each power of any quadratic form in three variables. Since the problem for rank $1$ and rank $2$ quadratic forms can be reduced to determining the rank of powers of binary forms, we primarily focus on non-degenerate quadratic forms. We begin by considering the quadratic form $q_{n}=x_1^{2}+\dots+x_n^{2}$ in an arbitrary number $n$ of variables. We determine the apolar ideal of any power $q_n^s$, proving that it corresponds to the homogeneous ideal generated by the harmonic polynomials of degree $s+1$. Using this result, we select a specific ideal contained in the apolar ideal for each power of a quadratic form in three variables, which, without loss of generality, we assume to be the form $q_3$. After verifying certain properties, we utilize the recent technique of border apolarity to establish that the border rank of any power $q_3^s$ is equal to the rank of its middle catalecticant matrix, namely $(s+1)(s+2)/2$.