How to construct a zero-knowledge proof of a number of the form $n=p^a q^b$
Actually, it is not necessary for the prover to show that "exactly half" of elements with Jacobi symbols being +1 are, in fact, QR.  Instead, here are some hints:


Assuming that n is not of that form (and also n is not of the form $n = p^a$, which is easy test for), the probability that a random element with Jacobi symbol +1 is a QR is at most $q$ (homework assignment for you: what is the value of $q$)
If we get a value $x$ from the verifier which is a QR, how can we show in a zero knowledge way that $x$ is a QR (with probability $> 1 - \delta$)?
If the provider gives us a series of random $x$'s, and we either respond to each $x$ with 'not a QR', or 'it's a QR; here's the zero knowledge proof with probability $> 1 - \delta$', how many trials with random elements would be needed before the verify can conclude (with probability $> 1 - \epsilon$) that the fraction of Jacobi symbol +1 elements which are QR are $&gt; q$?