  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 $(s+1)(s+2)/2$.

Algebraic Geometry