This reminds me of homomorphic encryption: you can perform computations on data while it remains encrypted, which would solve the trust problems you mention afaict. This seems like it would have enormous potential to change the cloud landscape. https://en.m.wikipedia.org/wiki/Homomorphic_encryption
My understanding (which is admittedly very little) though is that homomorphic encryption requires many times more computations compared to running on unencrypted data, so that could dampen it's usefulness.