How can create a proof that we have a sequence of binary digits? Given a sequence of bits (either a 0 or 1), for example:

0110100110110001


How can one create a proof that we have this bit sequence, without actually revealing the bit sequence?

I know that there are several examples of proof mediums, such as polynomials for zk-SNARKs, however, how would one go about defining a proof medium for the above question?

EDIT: This bit sequence is derived from some input x which passes through some arithmetic circuit C in which a set of environmental conditions (i.e. delay) determine the propagation through paths, such that (for example) the first path to the goal will be associated with a 1, else 0, which ultimately make up a sequence of bits.
If I understand correctly, you're asking to prove that you have the result of running a secret input through a public process without revealing anything about this result.

For this proof to be an interesting one, you would have to output some value and then prove how it relates to the values you're hiding. If you don't do that, then you're simply proving that you ran any value through a public process. Since anybody can run values through a public process, this wouldn't be interesting. Does that make sense?

In other words, it's important that you figure out what is interesting about your proof. i.e. is there some fixed secret information you would need to have to prove what you're trying to prove?

If you reveal, say, the encryption $c$ of the bit string under a public key $pk$. Then proving that $c$ is the encryption under $pk$ of some bitstring $y$ seems like it could be interesting. You would probably also have to prove that $pk$ was generated honestly. Assuming unique decryption, this combination of proofs will fix the witness $(x,sk)$ which leads to the output $(c,pk)$.

This proof can be framed as an NP language where the value $(c,pk)$ has witness $(x,sk)$. You can then use standard zero-knowledge.