We prove that for some constant $a > 1$, for all $k \leq a$,
$$\mathbf{MATIME}[n^{k + o(1)}] / 1 \not \subset \mathbf{SIZE}[O(n^{k})],$$
for some specific $o(1)$ function. Previously, Santhanam showed that there exists a constant $c>1$ such that for all $k > 1$:
$$\mathbf{MATIME}[n^{c k}] / 1 \not \subset \mathbf{SIZE}[O(n^{k})].$$
Inherently to Santhanam's proof, $c$ is a large constant and there is no upper bound on $c$. Using ideas from Murray and Williams, for all $k > 1$:
$$\mathbf{MATIME}[n^{10 k^2}] / 1 \not \subset \mathbf{SIZE}[O(n^{k})].$$

Our proof uses a new, very efficient $\mathbf{PCP}$ for $\mathbf{PSPACE}$. We construct a $\mathbf{PCP}$ for $\mathbf{SPACE}[O(n)]$ that has a $\tilde{O}(n)$ time verifier, $\tilde{O}(n)$ space prover, $O(\log(n))$ queries, and polynomial alphabet size. Prior to this work, $\mathbf{PCP}$s for $\mathbf{SPACE}[O(n)]$ either used $\Omega(n)$ queries or had verifiers that run in $\Omega(n^2)$ time.

1. Revised and simplified introduction.
2. Added some more notes about limitations of this result.
3. Added a new section outlining a strengthening of this result to OMA versus BPP/poly.
4. Other minor grammar fixes and slight changes to the PCP proof.

We prove for some constant $a > 1$, for all $k \leq a$,
$$\mathbf{MATIME}[n^{k + o(1)}] / 1 \not \subset \mathbf{SIZE}[O(n^{k})],$$
for some specific $o(1)$ function. This improves on the Santhanam lower bound, which says there exists constant $c$ such that for all $k > 1$:
$$\mathbf{MATIME}[n^{c k}] / 1 \not \subset \mathbf{SIZE}[O(n^{k})].$$
There is inherently no upper bound on $c$ in Santhanam's proof. Using ideas from Murray and Williams, all $k > 1$:
$$\mathbf{MATIME}[n^{10 k^2}] / 1 \not \subset \mathbf{SIZE}[O(n^{k})].$$

Our proof uses a new, very efficient $\mathbf{PCP}$ for $\mathbf{PSPACE}$. We construct a $\mathbf{PCP}$ with unbounded proof length for $\mathbf{SPACE}[O(n)]$ that has a $\tilde{O}(n)$ time verifier, $\tilde{O}(n)$ space prover, $O(\log(n))$ queries, and polynomial alphabet size. Prior to this work, $\mathbf{PCP}$s for $\mathbf{SPACE}[O(n)]$ either used $\Omega(n)$ queries or had verifiers that run in $\Omega(n^2)$ time.