PRF from hash function? Let $G=\langle g \rangle$ be a cyclic group of prime order $Q$. Let $H_1:\{0,1\}^*\to G$ be a hash function. Is the following a family of PRF for $s \in \mathbb{Z}_Q^*$?

$$f_s(x)=(H_1(x))^s$$

Or equivalently, let $H_2:\{0,1\}^*\to \mathbb{Z}^*_Q$ be a hash function,

$$f_s(x)=(g^{H_2(x)})^s$$

Intuitively it is in ROM, but can somebody prove it?
The first construction is a well-known PRF in the random oracle model, with security from DDH. I can't find a good reference right now where this construction is explicitly framed as a PRF, but I will update if I find it.

Edit: this is the PRF that is implicit in private set intersection protocols dating back to:
Catherine Meadows. A More Efficient Cryptographic Matchmaking Protocol for Use in the Absence of a Continuously Available Third Party. In IEEE Symposium on Security and Privacy. 1986
Bernardo A. Huberman, Matt Franklin, and Tad Hogg. Enhancing Privacy and Trust in Electronic Communities. In ACM Conference on Electronic Commerce. 1999

The second construction is insecure. If I know $f_s(x)$, then I can raise this value to the $H_2(y)/H_2(x)$ power (inversion is mod $Q$) and this tells me $f_s(y)$. So I can easily distinguish $f_s(y)$ from random.
BTW, in the random oracle model, $f_s(x) = H(s,x)$ is a PRF, as is $f_s(x) = g^{H(s,x)}$.