Is the BB84 protocol an example of "quantum supremacy"?

This is a fairly broad question, I hope it fits here. I am wondering if the BB84 protocol is an example of "quantum supremacy", ie. something a quantum computer can do but something that is assumed a classical computer cannot do the equivalent of. Are there any classical algorithms that can allow for the equivalent of secure key distribution as accomplished through BB84, with unconditional security proof? If not, what would it mean if there did exist such protocol?

That is, a key aspect of a programmable quantum computer is the ability to fully explore much of the Hilbert space afforded it. This means that a programmable quantum computer having $$n$$ qubits can prepare the qubits into a superposition in a Hilbert space of dimension $$2^n$$ - e.g., exponential in the number of qubits.
Turning to BB84, the BB84 scheme in quantum cryptography does not use any entanglement, and Alice and Bob (and Eve) only work on product states of the $$n$$ photons traded therebetween. The dimension of their Hilbert space only grows linearly with the length of the secret key. There is no computation in the BB84 quantum key exchange. Similarly, although E92 uses entangled Bell pairs, the Hilbert space dimension of E92 still only grows linearly with the number of qubits exchanged.