Is spectrum of Hamiltonian all you need? This should be well-known, but I don't seem to know it...
Quantum mechanics is defined by a Hamiltonian, and a Hamiltonian (as any Hermitian operator) is determined by its spectrum. Hence, it seems as if a spectrum determines a quantum theory.  I can see how this might work for a harmonic oscillator or some other simple systems, but if someone hands you say the Standard Model spectrum, can you really get anything from it? Where would you even start? Can you even get Poincare invariance, or even the concepts of momentum/position, starting from just a spectrum?
 A: I'll start on a soapbox, but then I'll step down and highlight a very interesting recent paper.
A quantum model is not defined by a Hamiltonian and a Hilbert space, despite what many carelessly-worded texts seem to say. To specify a model, we need to specify its observables — which operators represent which measurable quantities. Many authors make the reader guess what a model's observables are, based on how the author constructed the Hamiltonian and Hilbert space. The "right" guesses may often be strongly suggested by a given construction, but making the reader guess is still bad practice.
Now I'll step down off my soapbox and call attention to the paper arXiv:1702.06142 (Locality from the Spectrum). Here's an excerpt from the abstract:

...the energy spectrum alone almost always encodes a unique description of local degrees of freedom when such a description exists, allowing one to explicitly identify local subsystems and how they interact. As a consequence, we can almost always write a Hamiltonian in its local presentation given only its spectrum.

This is what they mean by "a Hamiltonian in its local presentation": Given a factorization of the Hilbert space into $n$ "subsystems" of equal size, a Hamiltonian with a "local presentation" (relative to those subsystems) can be written as a sum of terms, each of which is a tensor product of no more than $k$ single-subsystem operators. In more detail, the authors ask two questions:

*

*For a given $k$, does a generic Hamiltonian have any $k$-local presentation? The answer is no. Most Hamiltonians do not. The authors demonstrate this using a dimension-counting argument. (Beware that strictly local QFTs don't always have strictly local lattice versions. See the paper for a speculative comment about approximate locality.)


*If a given Hamiltonian does have a $k$-local presentation, then is that presentation unique? They define "unique" modulo various natural equivalences whose details I won't repeat here. The authors don't quite answer this uniqueness question directly, but they do derive a closely related result: If there is a single example of a $k$-local Hamiltonian on $n$ subsystems of the given size whose $k$-local presentation is unique (modulo natural equivalences), then almost all $k$-local Hamiltonians on $n$ subsystems of the given size also have unique $k$-local presentations (again modulo natural equivalences).
The authors assume a finite-dimensional Hilbert space, but this isn't necessarily a severe restriction in practice. In the authors' words:

All results presented are derived for models with a finite number of finite-dimensional subsystems. Such models may be used to approximate regularized quantum field theories... Interesting subtleties may exist for infinite-dimensional systems... However, we speculate that results of a similar spirit would still hold in the large-system limit.

These results don't contradict my soapbox-paragraph at the beginning of this answer, but they do illustrate in a surprising way that "the 'right' guesses may often be strongly suggested" even if the Hamiltonian is the only observable that is explicitly specified.
