Is it possible to use some cryptographic scheme to prove bitcoin ownership without revealing any information about value held in addresses?
Dagher et al. Provisions privacy-preserving proofs of solvency may be what you are looking for:
The objective of Provisions is to enable an exchange E to publicly prove that it owns enough bitcoin to cover all its customers’ balances such that (1) all customer accounts remain fully confidential, (2) no account contains a negative balance, (3) the exchange does not reveal its total liabilities or total assets, and (4) the exchange does not reveal its Bitcoin addresses.
Full version of the paper available in J. Bonneau's web page.