  We prove that the blocklength $n$ of a linear $3$-query locally correctable
code (LCC) $\mathcal{L} \colon {\mathbb F}^k \to {\mathbb F}^n$ with distance
$\delta$ must be at least $n \geq 2^{\Omega\left(\left(\frac{\delta^2
k}{(|{\mathbb F}|-1)^2}\right)^{1/8}\right)}$. In particular, the blocklength
of a linear $3$-query LCC with constant distance over any small field grows
exponentially with $k$. This improves on the best prior lower bound of $n \geq
\tilde{\Omega}(k^3)$ [AGKM23], which holds even for the weaker setting of
$3$-query locally decodable codes (LDCs), and comes close to matching the
best-known construction of $3$-query LCCs based on binary Reed-Muller codes,
which achieve $n \leq 2^{O(k^{1/2})}$. Because there is a $3$-query LDC with a
strictly subexponential blocklength [Yek08, Efr09], as a corollary we obtain
the first strong separation between $q$-query LCCs and LDCs for any constant $q
\geq 3$.
  Our proof is based on a new upgrade of the method of spectral refutations via
Kikuchi matrices developed in recent works [GKM22, HKM23, AGKM23] that reduces
establishing (non-)existence of combinatorial objects to proving
unsatisfiability of associated XOR instances. Our key conceptual idea is to
apply this method with XOR instances obtained via long-chain derivations, a
structured variant of low-width resolution for XOR formulas from proof
complexity [Gri01, Sch08].
