Before talking about upper bounds for $\mathsf{QMA(k)}$, let us first concentrate on upper bounds for $\mathsf{QMA}$.
Recall that the maximum acceptance probability of a $\mathsf{QMA}$ verifier $V_x$ is $\max_{|\psi\rangle} \||1\rangle\langle 1|_{out} V_x |\psi\rangle |\bar{0}\rangle\|_2^2$ where $|\bar{0}\rangle$ are ancillary qubits. It is a quadratic form, so $p_{acc}(\psi) = \langle \psi| M_x |\psi\rangle$ where $M_x := \langle \bar{0}| V_x^{\dagger} |1\rangle\langle 1|_{out} V_x |\bar{0}\rangle$.
Then we could show a series of upper bounds of $\mathsf{QMA}$:
- $\mathsf{NEXP}$: Note that $|\psi\rangle$ consists of exponentially many parameters, also we only need to distinguish $p_{acc} \geq c$ from $p_{acc} \leq s$ where the completeness-soundness gap $c-s$ is at least inverse polynomial. We can simply guess a $|\psi\rangle$, and calculate $\langle \psi| M_x |\psi\rangle$ by an $\mathsf{EXP}$ machine.
- $\mathsf{EXP}$: Notice the maximum acceptance probability of a $\mathsf{QMA}$ verifier $V_x$ is equal to the maximum eigenvalue of the associated matrix $M_x$. We can find the maximum eigenvalue of the exponential-size matrix $M_x$ by some convex optimization algorithm, which polynomially depends on the size of $M_x$.
- $\mathsf{PSPACE}$: By Prop. 14.5 in [KSV02], the maximum eigenvalue of $M_x$ can be well-approximated by a few linear algebraic operations, which can be done in $\mathsf{NC}(poly)=\mathsf{PSPACE}$.
The upper bound for $\mathsf{QMA}$ could be further improved to $\mathsf{PP}$ [MW05] (or slightly stronger, namely, $\mathsf{A_0PP}$ [Vyalyi03]).
Now let us move back to $\mathsf{QMA(k)}$: merely the first $\mathsf{NEXP}$ upper bound works for $\mathsf{QMA(k)}$, and all other uppers are unfortunately failed. The reason why is the maximum acceptance probability of a $\mathsf{QMA(k)}$ verifier regards all $k$-product states, so maximizing $p_{acc}(\psi)$ is no longer a convex optimization problem.
Still, when people first thinking about $\mathsf{QMA(k)}$, namely in [KMY03], they show that $\mathsf{QMA(k)}$ with perfect soundness will collapse to $\mathsf{QMA}$ in the same condition (which is equivalent to $\mathsf{NQP}$).
Furthermore, [KMY03] is sort of proving that $\mathsf{QMA(k)} = \mathsf{QMA(2)}$, just the soundness parameter differs from the standard scenario. In fact, if we could do error reduction $\mathsf{QMA(k)}$ (they improve to $\mathsf{QMA(2)}$ in the journal version), then [KMY03] will imply the actual equivalence. It is worthwhile to mention that the standard approach (i.e., do parallel repetition and accept if at least $\alpha$ ratio of copies is accepted) for error reduction only works in the completeness error for $\mathsf{QMA(k)}$. Few years later, [HM10] points out $\mathsf{QMA(2)}$ can be strengthened a bit, namely $\mathsf{QMA(2)}=\mathsf{QMA^{SEP}(2)}$, where $\mathsf{QMA^{SEP}(2)}$ means that the equivalent measurement $\{V_x^{\dagger} |0\rangle\langle 0|_{out} V_x, V_x^{\dagger} |1\rangle\langle 1|_{out} V_x\}$ can be further restricted to be separable (regarding mixed states). It turns out proving that $\mathsf{QMA(k)} = \mathsf{QMA(2)}$ since reducing soundness error is easy for $\mathsf{QMA^{SEP}(2)}$!
There are still a few more to say about upper bounds for $\mathsf{QMA(2)}$. There is a very early paper [KM03] claiming that $\mathsf{QMA(k)}$ with perfect completeness is upper bounded by $\mathsf{EXP}$. The proof outline is based on a new complete problem for $\mathsf{QMA(2)}$ and a deterministic exponential-time algorithm for it, but there is no further detail. There is also a failed attempt for an $\mathsf{EXP}$ upper bound [Schwarz15] several years ago. Beyond that, [ABDFS08] and [GSSSY17] provide a few conditional upper bounds, such as $\mathsf{PSPACE}$ or the counting hierarchy (which is contained in $\mathsf{PSPACE}$), based on various assumptions.
