  We study the logical complexity of proofs in cyclic arithmetic
($\mathsf{CA}$), as introduced in Simpson '17, in terms of quantifier
alternations of formulae occurring. Writing $C\Sigma_n$ for (the logical
consequences of) cyclic proofs containing only $\Sigma_n$ formulae, our main
result is that $I\Sigma_{n+1}$ and $C\Sigma_n$ prove the same $\Pi_{n+1}$
theorems, for all $n\geq 0$. Furthermore, due to the 'uniformity' of our
method, we also show that $\mathsf{CA}$ and Peano Arithmetic ($\mathsf{PA}$)
proofs of the same theorem differ only exponentially in size.
  The inclusion $I\Sigma_{n+1} \subseteq C\Sigma_n$ is obtained by proof
theoretic techniques, relying on normal forms and structural manipulations of
$\mathsf{PA}$ proofs. It improves upon the natural result that $I\Sigma_n$ is
contained in $C\Sigma_n$. The converse inclusion, $C\Sigma_n \subseteq
I\Sigma_{n+1}$, is obtained by calibrating the approach of Simpson '17 with
recent results on the reverse mathematics of B\"uchi's theorem in
Ko{\l}odziejczyk, Michalewski, Pradic & Skrzypczak '16 (KMPS'16), and
specialising to the case of cyclic proofs. These results improve upon the
bounds on proof complexity and logical complexity implicit in Simpson '17 and
also an alternative approach due to Berardi & Tatsuta '17.
  The uniformity of our method also allows us to recover a metamathematical
account of fragments of $\mathsf{CA}$; in particular we show that, for $n\geq
0$, the consistency of $C\Sigma_n$ is provable in $I\Sigma_{n+2}$ but not
$I\Sigma_{n+1}$. As a result, we show that certain versions of McNaughton's
theorem (the determinisation of $\omega$-word automata) are not provable in
$\mathsf{RCA}_0$, partially resolving an open problem from KMPS '16.
