Can keys from Bitcoin's Hierarchical Deterministic Wallets be correlated (reducing privacy)? I'm trying to understand if the feature "Hierarchical Deterministic Wallets" in Bitcoin allows for complete privacy of all derived keys, and if any of those keys can be associated with each other after they are generated.

Knowing the answer to this will greatly impact the scenarios these keys are usable in.
I recently came across a paper that may interest you that I think answers your question. To quote from the abstract:


  Unfortunately, in all existing HD wallets---including BIP32 wallets---an attacker can easily recover the master private key given the master public key and any child private key. This vulnerability precludes use cases such as a combined treasurer-auditor, and some in the Bitcoin community have suspected that this vulnerability cannot be avoided.


Combine this with the fact that child private keys are generated from the master private key, and it appears the answer to your question is no, they do not offer complete privacy.