Efficiently deciding whether TM accepts all inputs in at most $k$ steps
Given a 3CNF formula on $n$ variables, we can construct in polynomial time a Turing machine with polynomially many states $a(n)$ that accepts a truth assignment and in polynomial time $b(n)$ checks whether it satisfies the formula, and if so, enters an infinite loop. If you believe ETH, then checking whether this machine halts on all inputs in $b(n)$ steps should take time $2^{\Omega(n)}$, indicating some kind of exponential blow-up.