Why is the conduction band state s-type and the corresponding valence band state p-type?
This is not generally true. What material are you considering and where in k-space?
You really should cite where you get such a thing.
As DrDu as pointed, this is not true always and it is material dependent. For example, Cu, a common metal, has the 3d electrons as its valence shell, and thus, the 3d electrons that participate in the conduction band, not the 4s electrons. This already negates the starting point of your question.
For example, here https://www3.nd.edu/~djena/kdotp.pdf [Broken] it is stated that "For direct-gap semiconductors, for states near the conduction-band minimum (k = 0), the Bloch lattice-function possesses the same symmetry properties as a |S> state . In other words, it has spherical symmetry. The states at the valence band maxima , on the other hand, have the symmetry of p-orbitals"
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